Incircles

Given triangle ABC with side lengths a, b, and c. Let D, E, and F be the points of tangency of the incircle, as shown.

(a) Prove that triangle DEF is acute, that is, that the triangle determined by the points of tangency of the incircle is always acute.

(b) Find the area of triangle DEF in terms of a, b, and c.

(c) Show that AD, BF, and CE are concurrent. Explore this point of concurrency as the shape of triangle ABC is varied.

Hints/Solution:

The triangle DEF is the Pedal Triangle when the Pedal Point is the incenter I.

Strategy: Consider the area of triangle DEF as the sum of the areas of triangles EIF, DIE, and FID.

HINT

Comment:

Compare with the Excircle Problems

Reference: For an extension and partial solution, see problem 4153 in School Science and Mathematics, October 1987 and October 1988 issues.

In Problem 4153, A(I) is the area of the triangle from the tangent points of the incircle, and A(E1), A(E2), and A(E3) are the areas of the three triangles from the tangent points of the three excircles.