Question

Suppose ABC is a triangle inscribed in a circle. The bisector of angle ABC intersects the circle again in point D. The tangent in point D the tangent in point D intersects the line BA and line BC in E and F respectively. Prove that angleEDA = angleFDC ​

Answer 1

Answer:

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Answer 2

Given: ABC is a triangle inscribed in a circle, tangent in point D the tangent in point D intersects the line BA and  line BC in E and F.

To find: Prove that angle EDA = angle FDC .

Solution:

• So to prove, angle EDA = angle FDC, we need to first prove congurent triangles, so

• In triangle ABC,

            BD bisects angle ABC.......... (given)

            ang ABD = ang CBD    .....................(by angle bisector property)......(I)

• Now, in question we have given that AD is a secant, and EF is a tangent, so it concludes that:

            ang EDA ≅ ang ABD

• Similarly,

            ang CDF ≅ ang CBD  ............(II)

• By the property angles in alternate segments are congruent.

• Similarly, we can prove that:

             ang EDC  ≅  ang CBD...............(III)

• So, by using I, II and III

Answer:

           angle EDA = angle FDC                  ............Hence proved