Mathematics
If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.
Answer
Let AD bisect BC and also the angle A.
In △ ABD and △ ACD,
⇒ BD = CD (Since, AD bisects BC)
⇒ AD = AD (Common side)
⇒ ∠BAD = ∠CAD (Since, AD bisects angle A)
∴ △ ABD ≅ △ ACD (By S.A.S. axiom)
We know that,
Corresponding sides of congruent triangle are equal.
∴ AB = AC
∴ ABC is an isosceles triangle.
Hence, proved that if the bisector of an angle of a triangle bisects the opposite side, the triangle is isosceles.
Related Questions
In isosceles triangle ABC, AB = AC. The side BA is produced to D such that BA = AD. Prove that : ∠BCD = 90°.
In a triangle ABC, AB = AC and ∠A = 36°. If the internal bisector of ∠C meets AB at point D, prove that AD = BC.
Prove that the bisectors of the base angles of an isosceles triangle are equal.
In the given figure, AB = AC and ∠DBC = ∠ECB = 90°.
Prove that :
(i) BD = CE
(ii) AD = AE
