Question

In △ABC (m∠C=90°), the points D and E are the points where the angle bisectors of ∠A and ∠B intersect respectively sides BC and AC. Point G∈ AB so that DG ⊥ AB and H∈ AB so that EH ⊥ AB. Prove that △CEH and △CDG are isosceles. (preferably in statement reason)

Answer 1

-30000000000000

Explanation:

Answer 2

Triangle ABC is a right angle at C. The angles bisectors of BAC and ABC meet BC and AC at D and E respectively. The points H and G are the feet of the perpendiculars from E and D to AB.

As AD is an angle bisector,

⇒ ∠CAD = ∠BAD

⇒ ∠CAD = ∠GAD ( ∵ ∠BAD = ∠GAD)

The two right triangles CAD and GAD ( ∵ m∠G = 90° = m∠C ) share a hypotenuse and have equal acute angles,

⇒ ΔCAD = ΔGAD

⇒ CD = DG

∴ ΔCDG is an isosceles triangle. (Proved)

Likewise,

As BE is an angle bisector,

⇒ ∠CBE = ∠ABE

⇒ ∠CBE = ∠HBE ( ∵ ∠ABE = ∠HBE)

The two right triangles CBE and HBE ( ∵ m∠H = 90° = m∠C ) share a hypotenuse and have equal acute angles,

⇒ ΔCBE = ΔHBE

⇒ CE = EH

∴ ΔCEH is an isosceles triangle. (Proved)