Problem 25: Incenters in a Right Triangle

Problem 25

Let ABC be aright triangle, where B = 90°. Let AD be the altitude from A, where D lies on BC. If P, Q, and R are the incenters of triangles ΔABC, ΔADB, and ΔADC respectively, then show that:

AP = QR

Here’s a simulation that demonstrates the problem.

Try dragging the points B and C. Is AP always equal to QR? How can we prove the same?

Solution

To solve this problem, you’ll need to know the following.

Incenter of a Triangle

Cosine Rule

[coming soon]

Pythagoras Theorem

I’m assuming that you know this one. Anyways, here’s a quick recap:

In a right triangle, the square of the hypotenuse equals the sum of the squares of its other two sides.

Conversely:

If the square of one side of a triangle equals the sum of the squares of its other two sides, then this triangle must be right-angled.

Here’s a little simulation that demonstrates the theorem.

You can drag the three vertices of the right triangle. Is PQ2 always equal to QR2 + RP2?

Problem Source: MathClub

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