Sam asked in Science & MathematicsMathematics · 4 years ago

# triangles ABC is inscribed in a circle and AB=AC the bisector of angle ABC meets the tangent from A at D. Prove that AD and BC are parallel?

and also triangles ADC is isocles

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• 4 years ago

Join AE to E the mid point of BC. Because AB = AC AE must also be perpendicular to BC. So it must pass through the center of the circumscribed circle of triangle ABC. As AD is tangent ∠EAD must be a right angle. We have already proved that ∠AEC is a right angle As these internal angles of BC and AD formed by transverse AE add to two right angles AD||BC. Hence proved.

∠DBC = ∠BDA (alternate angles)

but ∠DBC = ∠DBA (given)

so ∠DBA = ∠BDA in the ΔABD, hence AB = AD but AB = AC (given so AC = AD hence ΔACD is isosceles..

• Anonymous
4 years ago

Due to have two of the same sides, the triangle has to be an isosceles.

The reason AD and BC are parallel as the bisector produces a line that is 90 degrees with the tangent line and with side BC