If triangle ABC is isosceles with AB=AC and C(O,r) is the incircle of the triangle ABC touching BC at L - Maths - Circles

Question

If triangle ABC is isosceles with AB=AC and C(O,r) is the
incircle of the triangle ABC touching BC at L ,prove that L bisects
BC

Mohil · Answer

ΔABC is an isosceles triangle C(O, r) is the incircle of ΔABC touching BC at I C(O, r) is the incircle of ΔABC ∴ O is the point of intersection of angle bisector of ΔABC I e , OB bisects ∠B and OC bisects ∠C In ΔABC, AB = AC (Given) ∴ ∠C = ∠B (Equal sides have equal angles opposite to them) $n n \Rightarrow n n \frac{n}{n 1 n} n â¢ n n n n n ∠ n n n C = n \frac{n}{n 1 n} n â¢ n n n n ∠ n n n B n n$ ⇒ ∠OCI = ∠OBI (OB bisects ∠B and OC bisects ∠C) InΔOBI and ΔOCI, ∠OIB = ∠OIC (Radius is perpendicular to the tangent at point of contact) ∠OBI = ∠OCI (Proved) OI = OI (Common) ∴ ΔOBI $n n ≅ n n$ ΔOCI (AAS concurrence criterion) ⇒ BI = IC (CPCT) Thus, I bisects the side BC

If triangle ABC is isosceles with AB=AC and C(O,r) is the incircle of the triangle ABC touching BC at L ,prove that L bisects BC