**Ternary diagrams** are common in **chemistry and geosciences** to display the **relationship of three variables**. Here is an easy step-by-step guide on how to plot a ternary diagram in Excel.

Although ternary diagrams or charts are not standard in Microsoft® Excel, there are, however, templates and Excel add-ons available to download from the internet. Nevertheless, it might be advantageous to build your own template as it gives you more freedom to display data and which additional parts (ticks, lines, values, etc.) you want to display. Find a step-by-step guide on how to generate a ternary chart in Excel that you can customize towards your requirements.

## How does a ternary diagram work?

A ternary or triangular diagram (chart/plot/graph) is a graphical way to display the relationship between three variables or components.

To display three variables in a ternary graph, their values must be normalized to a constant, usually either 1 or 100%.

The advantage of ternary plots is that three variables can be displayed in a two-dimensional way. The disadvantages are that these variables are normalized to a constant (1 or 100%) and therefore have different values than initially, which are also not independent of each other.

I assume the data will be normalized to 100% in the following descriptions, but the theory is the same for normalization to 1%. By the way, all ternary plots here are generated in the way I am going to show you.

I follow clockwise labeling of the ternary diagram, i.e., A on top, B on the right (bottom), and C on the left (bottom apices.

Before we start, just a quick explanation of how to read a ternary diagram helping us understand what we need to achieve.

### How to read a ternary diagram?

A ternary diagram consists of an equilateral triangle of which the apices represent 100% and the opposite base/side 0% of each variable.

Let’s assume you have three variables, A = 42.5%, B = 25.5%, and C = 17%. This sums up to 85%. Those values change to A = 50%, B = 30%, and C = 20% when normalized to 100% (see math/formula below).

Plotted into a ternary graph, it looks like this (Figure 1):

Let’s start with component A: the apex A (top) represents 100%, and the base opposite the apex equals 0%. So starting from the base towards the top apex, we can determine the value of component A (Figure 2), here 50%.

Now, we do the same with component B (Figure 3) starting from the base (0%) opposite apex B on the lower right (100%), which gives us the value of 30%.

And finally, component C is determining from the right side/base (0%) towards apex C on the bottom left side leading to a value of 20% (Figure 3).

## Building a ternary diagram in Excel

Microsoft® Excel displays two axes only. The trick is thus to convert/project our ternary system into a binary system. This sounds more complicated than it is.

### STEP 1: Preparing the spreadsheet

In the example I use here, I am using 3 columns for the original (non-normalized) geochemical data i.e., 3 components. Then another 3 columns for the normalized values, and a fourth column for the sum of the previous three as a check option. This is followed by two columns, which will show the XY coordinates of the binary plot of the ternary diagram.

In Figure 4, you can see the basic outline of the spreadsheet.

#### Normalizing the original data

Like mentioned before, the data we want to display in a ternary diagram need to be normalized to either 1 or 100%; here we are using the 100% approach.

We start with determining the coordinates for our ternary diagram apecis, i.e., the outline of it (triangle); i.e., A = 100%, B = 100%, and C = 100%.

In the spreadsheet, we type 100 into cell B4, 0 into cell C4, and 0 into cell D4 (meaning 100% A = A-apex). For the B-apex, we type 0 into cell B5, 100 into cell C5, and 0 into cell D5, and finally for the C-apex 0 into cells B6 and C6, and 100 into cell D6 (see Figure 5).

Next, we set up the cells and columns to normalize the initial values in columns F, G, and H.

To normalize component A to 100% in relation to the other components use the following formula:

According to this, type the following into cell F4 to normalize A’ to 100%: =100*B4/(B4+C4+D4), see the Excel Formula Bar in Figure 5. Cell G5 is =100*C4/(B4+C4+D4) and cell H5 gets =100*D4/(B4+C4+D4) to normalize components B’ and C’ respectively.

I put the sum of columns F, G, and H into column I (i.e., =SUM(F4:H4) or =F4+G4+H4), for double-chicking everything works fine. Then drag the content of cells F4 to I4 down. Your spreadsheet should have the numbers in it like under the blue and green marked cells in Figure 5.

You may argue that the initial data (A’, B’, and C’) are already summing up (are normalized) to 100%. In this case, that is right. You can test, however, if your cells are set up correctly, by replacing the 100 entries with random numbers, the normalized values should stay the same. Double-check by additionally replacing a 0 in the non-normalized data with a random number too. The normalized value should change accordingly, but the sum should still be 100 (Figure 6).

#### Calculating the X and Y coordinates

The next step is to calculate the X and Y coordinates so we can plot the triangular (ternary diagram).

I set up columns K and L for the X and Y coordinates (Figures 4 and 5).

The conversion into X and Y coordinates is straightforward:

#### The XY coordinates of the triangle

Putting this into action, we know the ABC coordinates of the triangular/ternary, and applying the conversion as outlined above, we get the XY coordinates as shown in Figure 5, columns K and L.

To close the triangle shape, we need to add one point twice, but in order (Figure 8)

Copy the values of the calculated XY coordinates to a separate place in the spreadsheet so they are out of the way (e.g., to columns AA and AB and label them accordingly, Figure 8).

We are now ready to plot the outline of the triangle, which will become our ternary diagram.

### STEP 2: Plotting the ternary into a binary diagram

#### Insert a Scatter Chart

Insert a Scatter Chart (XY diagram), e.g., ‘Scatter with Straight Lines’ (Figure 9) using the XY coordinates for the triangle from columns AA and AB. To make it into an equilateral triangle resize the chart area accordingly; for example 10 columns wide and 30 rows high, as in Figure 10. (You can check by drawing a triangle (Insert > Shapes > Triangle); hold the Shift key, which will make the triangle equilateral, while drawing the triangle over the diagram.)

#### Adjust the XY axes scales

Excel will automatically scale the X and Y axis from 0 to 120; we need to change both axes to 0 to 100 (see Figure 11).

#### Adding labels to the apices

Next, we need some space for the apices labels: click into the Plot Area (not the Chart Area) then resize by holding the Shift key (this ensures an equal scaling) and use the mouse cursor on one of the corner pick-points. Then recentre the Plot Area in the Chart Area.

Insert a Text Box (Insert > Shapes > Text Box) on top of the apex A, and instead of text type in (or use the mouse and click into the cell) the cell location. In our example the text box for the top apex “=$F$3”, right bottom “=$G$3”, and left bottom “=$H$3” (displaying A (centered), B (aligned right), and C (aligned left), respectively); Figure 12. In this way, if you change the entry of the labels in cells F3 (A), G3 (B), and H3 (C), it will automatically show in the diagram.

#### Cleaning up the chart

Delete the vertical and horizontal gridlines.

Delete the Y-axis.

Change the line color of the triangle (if you like) and adjust the line thickness, i.e., same for triangle and X-axis.

#### Adding tick marks and labels to the axes

##### Adding tick marks

We start with the X-axis; like in an XY chart, add tick marks to the X-axis (recommended type: *Cross* rather in *Inside* or *Outside*; see below).

Add two new data tables with coordinates and labels, as in Figure 13, to your Excel spreadsheet, e.g., close to the coordinates for the triangle, and somewhat out of the way.

Add the XY coordinates as new data series to the diagram (Figure 14).

The data points will likely be displayed as a line. Change their appearance from a ‘line’ (*Format Data Series* > *Fill and Line* > *Line*) (Figure 15) to ‘long dash’ symbols (Figure 16); the long dash will mimic the *Cross* type of tick marks of the X-axis set up earlier. Change the color accordingly to those of the X-axis.

Do this for the axis between A and B, and between A and C.

##### Adding labels to the tick marks

Use the* Add Chart Element* > *Add Labels* (*Chart Design tab*) to add *Data Labels* to the A to B axis aligned to the right (Figure 17), then add *Data Labels* aligned left to the C to A axes.

By default, Excel will use the *Y Value* as Data Label. You can change this, if you want, for instance, the labels in the opposite direction, by changing the Label Option (Format Data Labels) to *Value From Cells*, and then use the *Select Range* button to select the cells (here the labels we wrote earlier).

#### Adding gridlines

In the final step, we add gridlines to the graph. We need to add another table with XY coordinates. You can calculate the coordinates with our template, but I provide the coordinate in a separate table [follow this link]. I copied the data, again somewhat out of the way, into columns AA and AB.

Add another data series to the graph with the coordinates for the gridlines. In our example, the data are displayed as dotted lines in grey.

## Step 3: Add your data

So, the spreadsheet and chart are set up and finally ready to display your data.

Add the raw (non-normalized) data into columns B, C, and D (starting with row 4). You may have to extend the range of columns F to I, K, and L to calculate all parameters (Figure 20).

Then add the new data points to the triangular diagram (e.g., right mouse click into chart > *Select Data* > Add New Data Series > *Edit Series*, and select the data in columns K and L), just as you usually do.

Note: Have you noticed that I added labels for the three apices into cells F3, G3, and H3, which then are displayed at the apices of the ternary diagram (Figure 20); – remember how we set this up in step 2.3 (Figure 12)

## How about displaying data point values?

It may be useful to display the actual ternary values next to the data points in the diagram. If you (right mouse click on data points > *Add Data Labels*), Excel will display by default the *Y-Value*, i.e., the values from column L. Double-click in the data labels and you can add the *X-Value* and number of digits to be displayed. This may be adequate for your purpose, e.g., identifying a certain data point in the table.

However, maybe you want to display the actual ternary coordinates of the data points. Here is how to do that:

#### Setting up the ternary coordinates display

To display the actual ternary coordinates (A, B, C) we need to do some preparational work.

In the example below (Figure 21), I extended column Y between the chart and the coordinates for the triangle, tick marks, for the data labels we want to display. You can, however, put them wherever you see it fits best for your purposes.

I called the heading ‘Data Labels’; note the heading is aligned with those from the data, i.e., row in 3 (Figure 22).

In row 4 (below that header), type in the following formula:

=CONCAT(TEXT(F4,”0.0″),”, “,TEXT(G4,”0.0″),”, “,TEXT(H4,”0.0”))

‘CONCAT’ is the Excel function combining text from several cells, ranges, and or strings (note: in older Excel versions the function is called ‘CONCATENATE’). The ‘TEXT’ function converts a numeric value to text, which enables us to give it a certain format. This format is expressed by the string “0.0”, which restricts the number format to one digit behind the comma (“0” would mean no digits, and “0.00” two digits behind the comma respectively). F4, G4, and H4 are the cells we want to combine; see Figure 22 for a visual explanation.

Coming back to displaying the data point labels in the chart. Like before, or your preferred way, add *Data Labels* to the data points and open the *Format Data Labels* dialog (e.g., right mouse click on data points > *Add Data Labels*, or if you have already data labels shown, double click on the labels), untick *Y Value* and/or *X value* and tick *Value From Cells*. Then use the *Select Range* button to select the cells in column Y; similar to Figure 18).

That’s it. You should have a ternary diagram plotted in Excel that looks similar to the one in Figure 23.

## Related article

The geochemical differentiation diagram for sedimentary rocks after Mason (1966)

Dear Christian,

I also want to say a big thank you for this very detailed explanation! It was so easy to follow.

Kind regards,

Kata

Thank you for your feedback Kata.

Glad to hear that it was easy to follow.

Best wishes,

Christian

Just wanted to thank you for sharing this – really awesome to be able to do this for soil texture dat in excel.

Many thanks

Thank you for the encouraging feedback.

good day sir, i got stuck in adding labels and apices,

the part where the textbox shouldn’t be a lateral A but a command “=$F$3” i tried entering this value but does work

Greetings Emmanuel.

The cell reference “=$F$3” obviously refers to that cell, i.e. F3.

1) Make sure that is the correct reference, e.g. to cell F3. Is your ‘A’ of the normalised data in F3 or somewhere else?

2) I am not sure if they changed something in newer Excel versions or if the Mac version works differently (do you have a Mac?). My Ternary works like described (initially done on Windows laptop), but if I insert a new textbook, it doesn’t work this way anymore (switched to Mac).

Try the following: a) insert the text box, b) while the text box is still selected, place the curser into the formula bar and type “=” then click into the cell (e.g. F3) and confirm/enter. In my case the cell address is now “=Sheet1!$F$3”.

I hope that works.

Got the idea using the trick Y equals A and X as 0.5xA + B. It is very helpful. Thanks a lot

Hello,

Glad to see that you solved it out by yourself. In one of the earlier comments/replies a more detailed explanation is given.

Do you know how you might make a heatmap of different results? Let’s say you have a data point for every coordinate in the ternary chart. Each point has a different value and you want to make the whole triangle a heat map corresponding to those values. Is there a way to do that?

Hello Ben,

I am not sure if heat maps in graphs can be generated with Excel. The only function I know of, is related to values in the cells of the spreadsheet itself (i.e., via Conditional Formatting).

The ternary diagram construction here is a kind of cheat, because it uses a common XY-diagram. Thus, even if a heat map could be generated, it would not be restricted to the ‘inner triangle’, and rather fill the whole XY-diagram area.

Hi, just wanted to say that this is incredibly awesome. Instructions were clear and easy to follow. Thank you!

Thank you Andrew. Glad to hear that it was useful and easy to understand for you.

Hello Christian S., thank you so much for helping us create a ternary diagram. How do we change the value of the axis, so we can zoom in on the area we want?

Hello Dema,

Thank you for reaching out, and my apologies for the late reply. With this version it is not straight forward to ‘zoom’ in. I assume you have data points that plot rather towards one corner or upper/lower half of the diagram. Interesting question and food for thought on how to tackle this in Excel.

Dear Christian, you are my hero of today! Thank you so much for this wonderful, clear explanation. Without you I wouldn’t have managed to create a ternary diagram in Excel!

Dear Jan,

Thank you for your kind words; happy that it helps you.

This was an extremely helpful article. I was looking for a way to create and customise Ternary plots for my thesis. Your article helped me understand how to make one by myself. This is invaluable, thanks a lot for this!

One minor issue, but this maybe due to some issues on my laptop but when I copy paste the gridline coordinates from your linked page, the lines do not plot when the coordinate columns are added as a series to the ternary diagram chart. However, manually copying out the values in the exact same format, works fine.

Again, thanks!

Hello Kinjal,

Thank you for getting in touch, and I am glad to read it is helpful for your thesis. I am not sure what the reason for the copy issue may be.

Best wished for you and good success with your thesis.

Thank you very much!!!

This was extremely helpful!

Glad to hear that you found it useful.

Very helpful. Thanks a lot.

Sorry for the next comment.

The first five ternary charts in your explanation have a mistake. On the top apex, the two meeting axes are at 100%. That should not be. Each 100% of each ax must be located in a different apex.

Thank you Antonio,

You are absolutely right. I will change the charts asap when finding some time.

These charts are from an earlier version.

After trying and trying again with countless templates downloaded from the internet, this tutorial was an absolute godsend for me – amazingly clear instructions and will be invaluable for my dissertation work.

Hi Oliver,

Thank you for your feedback. I an happy and glad that my tutorial has such a good use for you.

All the best and good luck with your dissertation,

Christian

Hi! Thanks for your time, this is really helpful. Altough I could make it through, there’s a problem with excel when I select the input data from the coordinates cells that contain the formulas to transform the normlaized values to coordinates (approx 8000 rows). It gets really slow and excel keeps crashing every time, even when I have not paste my data into ABC columns yet. It seem that reading many cells consumes many resources, even when there are no values to plot (just tried with adding two row with values). However, if you have few data there would be no problem I guess. This is not my case, though.

Hello,

Thank you for your inquiry. The instability and slow response might indeed be due to the large number of cells you are using. I usually only prepare spreadsheets for maybe 250 data lines, or only for the number of actual required lines/rows. Have you tried the limit the lines (rows) to a couple of hundreds to see if Excel is more stabil then?

There might be a more elegant way in using the formulas, e.g. written in VBA scripts. That, however, is beyond my Excel capabilities.

Hello there,

Thanks for explaining this in really easy to follow directions. However, I am unsure of how I add my data without it changing the shape of the actual triangle. When I put my data in the raw data table the diagram and is calculated into the Diagram Coordinate table it changes the shape of the triangle. Sorry for silly question but not sure where I am going wrong.

Cheers,

Tim

Hello Tim,

Thank you for reaching out. There are no silly questions here; — others may have the same or similar issues when building the spreadsheet.

Answer:

Tim and I got together via email and sorted the problem. It was due to some cells not linked together in the correct way.

I hope the spreadsheet and ternary diagram work fine now.

Thanks very much, I made a very beautiful diagram, but I had a problem with adding my data values inside the diagram, I have three components in my system, but the percentage of the two interesting components is less than 14%, and the variation of these two values affect the final result

Reply Christian:We figured out the missing thing outside of this comments section. The diagram works now and I hope it is of good use.Thank you so much for posting this. I am able to write an AFM diagram for my data!

Thank you for your feedback; – glad to hear that it is useful and helped you building your AFM diagram.

Hello,

When you add markers, it seems you don’t mention how to add those from C to B axis (bottom triangle base). That would mean adding another xy coordinates tab, but the problem is that I don’t know how to rotate markers (horizontal markers on the oblique sides can be seen, but on the bottom side you can’t see them).

Hello,

Thank you for your questions. It always helps to see where I can improve the description, as I am sure you are not the only one struggling with this.

The bottom axis, i.e. C-B is the X-axis of the the graph. Remember, we ‘project’ a ternary diagram onto an XY-diagram/graph. In Figure 12 you see that both, X and Y axes are visible. The next step is to delete the Y-axis (“Cleaning up the Chart”). The X-axis is still visible. Here you can format the style of the ticks and numbers like in a ‘common’ XY-graph. For instance I chose ‘Cross’, which means the tick marks cross the axis and you have the mark inside and outside. Thereafter (“Adding tick marks and …”), we simply mimic that style graphically for the other two axes. As I understand you do not have problems with those steps.

I hope that explains it and helps you to construct your own ternary diagram.

(PS: I am planning to make the Excel spreadsheet available for downloading, but can currently not find the time to figure out how to do it. It will come someday. However, I hope there is also some fun involved in building it by your own, and of course some learning is always good.)

Hi, great explanation, can you share your Excel spreadsheet with the example? Many thanks in advance

Hi Oscar,

Thanks for reaching out. I thought about making the Excel spreadsheet available for download, but then decided against it. The reason was that it should be an educational exercise. I believe that when people build the sheet for themselves, they understand how it works, and thus they can implement changes and their own requirements, such as field boundaries, trendiness etc., easier.

However, I may do a download page soon. First I need to find some time and finalise some articles that are crucial for the subject (chemostratigraphy) of the website. After that I will look into how to generate downloads from a webpage.

I will keep you in the loop once that is live.

Hello Christian,

Thank you for answering my question. I have one more that just came up.

I have finished making the diagram and all my points are in the plot. But I need to join all of them together and I don’t know how to do that. Please help me out.

Thank you so much. I really appreciate your help.

This is the reply I wrote to Mark when his question came up a couple of weeks ago. We sorted it per email, but but I post the answer here again, so others may benefit from it:

Option 1:

1) When you select the data for the diagram, mark the first set, then hold the Ctrl key and mark the second set. Thus they will be treated as one set and can be connected (‘Scatter with straight line and marker’ or are you using trendlines? Doesn’t matter, same principle).

2) Add another ‘Data Series’ with just one of the two data sets and display in the diagram, too. If you want those points only as symbols display them as ‘Scatter’ (i.e. without line). They should plot over / overlay the combined set, and thus you can differentiate them, e.g., with a different symbol or colour.

Option 2:

1) Copy both data sets together, i.e. into one set. Then display/connect them.

2) Same as step 2 above.

(Alternatively to step 2, you can put the two sets together as per step 1 (either option 1 or 2), and then mark and change the appearance of the data points from one set manually. For this you need to click on the data point, thus the whole series will be selected, and then click on it again, now only that point is selected and can be edited, e.g. change symbol, colour, etc. However, that’s a bit work intensive, because it needs to be done for each of the points in that set.)

Hopefully that helps.

Thank you for writing this article.

I just had one question. For the co-ordinates for the gridlines you said ” [follow this link].” to open up another table, But there is no hyperlink. Could you please help me out.

Thank you once again

Hello Mark,

Thank you for reaching out, and my apologies for the missing link. I just reactivated it and it should work now.

Otherwise, you can go directly to https://chemostratigraphy.com/addons-for-plotting-a-ternary-diagram-in-excel/

I hope you find the instructions easy to follow and that the ternary diagram is of good use for you.

Hi there. Thank you for the article

I just have one question. For the gridlines section you said. ” You can calculate the coordinates with our template, but I provide the coordinate in a separate table [follow this link]. I copied the data, again somewhat out of the way, into columns AA and AB.” But I don’t think the hyperlink works. For the follow this link bit. Could you please tell me the co-ordinates for the gridlines?

Thank you so much. Really appreciate your help.

Hi Leif,

Thank you for reaching out. My apologies for the missing link. I just reinstated it and it should work now.

Otherwise, you can go directly to https://chemostratigraphy.com/addons-for-plotting-a-ternary-diagram-in-excel/

I hope you find the instructions easy to follow and that the ternary diagram is of good use for you.

First, many many thanks for the great effort you made here. It is very clear, and straightforward.

However, I have just one point that I really appreciate if you could you please elaborate more on it. That is how to insert the data in step# 3 (Add your data). I did everything correctly. However, step#3 (how to add the data of A,B,C) is not very clear to me. We only have x, y data to select. How to selected the third column as well and show it on the ternary diagram?

You said: “Then add the new data points to the triangular diagram (e.g., right mouse click into chart > Select Data > Add New Data Series > Edit Series, and select the data in columns K and L), just as you usually do”.

How to go from add new data series to edit series at the same time. Thanks and I really appreciate your help with that.

Thank you for getting in touch, Ahmed. No problem; here is a more detailed workflow for Step 3:

1) You copy your data (non-normalized) to columns B, C, and D starting with row 4, i.e., below the blue headers (RAW DATA A’, B’, and C’).

2) You probably have to extend the range of columns F to I, and K and L, as they only have formulas down to row 10. Simply mark the last cells (i.e., F10 to L10) and then drag the little square at the lower right corner of the marked cells (your mouse cursor changes to a + when on it) down to where you need it. Alternatively, you can copy cells F10 to L10 and paste them into the cells below for the extent you need).

3) Now right-click into the diagram (a small window opens) and click on ‘Select Data’.

4) A window opens called ‘Select Data Source’. Click the ‘Add’ button.

5) A window opens called ‘Edit Series’ (I think here is the confusion in my description with “… >Edit Series”).

6) You can give the series a name and select the Series X values (which will be from column K) and the Series Y values (here from column L). Then click ‘OK’, and ‘Ok’ again on the ‘Select Data Source’ window.

The data should now be displayed in the ternary diagram, which is based on X and Y values only, as we interpolate the three components onto a binary diagram.

I hope this helps to understand the process a bit better. If you have further questions, I can send you Screen Shots.

Hello! I appreciate very much your effort into teaching us who to built our own diagram. However i’m still struggling on step #3. I get it that the Y coordinate was the first column of your spread sheet (SiO2), but I didn’t understand the results for the X coordinate. Can you elucidate that for me?? I still didn’t get it from the answer you gave to other reply.

Thank you very much, directly from Brazil 🙂

Hello Nathana, thank you for getting in touch. The ternary diagram in Excel is a projection of the three components (A, B, C) onto a binary (X, Y) graph. Therefore, the three components have to be converted into two. As per the second part in Step #1, Y equals A (in the example SiO2), and X is calculated as 0.5xA + B. The factor 0.5 comes from a trigonomic function (tan); the math behind it is explained in the discussion with Bob Mansfield (the first comment on this page). It is a bit complicated, as you can see from that reply. To demonstrate that you may simply want to take a ruler and measure the length of X (e.g. MgO to CaO) and the hight of Y (e.g. SIO2). Thus the 100 units (0-100%) of X (MgO to CaO) and that of Y are in a relation of 100:86.600254. This is compensated in with the factor 0.5 instead of 0.57735 (see discussion with Bob). I hope this answers your question (?).

Now I got it!! Thank you very much for taking your time explaining this for me!!

You’re welcome!

“The X-axis equals half the height of component A plus component B, thus X = 0.5xA + B.”

Using 0.5 in the statement above is imprecise and will lead to imprecise results. The correct value is 0.57735 (Tan 30 degrees) and will give much better results.

Thank you for your valid comment. You are right with the tan(30°) statement for calculating the X coordinate in an equilateral triangle/ternary diagram. In the case here, however, it is a different approach:

The ternary diagram ABC is projected into a cartesian XY graph.

For an equilateral triangle with a side length of 100 units (e.g., X-axis; B-C side 0-100%), the height is 86.60254 units (i.e., Y-axis; A).

A is projected onto Y with 100 units. Thus, we can use the normalized A component as Y without a trigonometric function (Fig. 7).

Because A is projected to 100 units (rather than 86.60…), the triangle is elongated and the angle of 30 degrees changes to 26.56505 degrees [tan(26.56…°) = 0.5]; unfortunately, Excel doesn’t plot diagrams in scale to show this. As per Step 2, the triangle/ternary diagram has then been stretched, so it looks like an equilateral triangle again, which doesn’t affect the XY coordinate system behind it, as it is stretched with it.

Correspondingly, due to the projection of A to Y (100 units), a factor of 0.86603 comes into play: tan(30°) = 0.57735; -> 0.57735 x 0.86603 = 0.5.

I also tested the ternary diagram, as constructed here, extensively, and the data points plot exactly where they should be (see for instance the first three diagrams (Fig. 1-3).