Geometry Problem 11: Equilateral Triangle

Problem 11

ABC is an equilateral triangle, insrcibed in a circle. If D be any point on the minor arc AB, then prove that:

AD + BD = CD

Here’s a simulation that demonstrates the problem.

Try dragging the point D, and observe the values of AD + BD and CD. Are they always equal?

Solution

Here’s what you’ll need to know to solve this problem.

Sine Rule

In any triangle ABC,

BC/sinA = CA/sinB = AB/sinC

Here’s a simulation where you can explore this property.

You can drag the three vertices and observe the three ratios BC/sinA, CA/sinB, and AB/sinC. Are they always equal?

Angles Subtended By a Chord

Try dragging the points A, B, C, and D (keeping them on the same side of PQ). You can also drag P and Q. Do the four angles always remain equal to each other?

Now, to proceed, you can apply the sine rule in triangles ADC and CDB, relating AD and BD with CD.

Then, see which angles are equal using the second theorem I mentioned.

Finally, using some trigonometric identities, some terms should cancel out and lead to the final result.

That’s all for now. I suggest you proceed on your own from this point. Good luck!

Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Facebook page.

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