1) In a triangle of ABC in which D is a point on side BC. Where \(BD = AD = DC\). Find the length of side AD if side \(AB = 15\,cm\) and \(AC = 20\,cm\).

2) In a triangle of ABC in which D is a point on side BC. Where \(BD = AD = DC\). Find the ∠B if ∠C = 55°.

3) In a triangle of ABC in which D is a point on side BC. Where \(BD = AD = DC\). Find the area of the circle which is inscribed in triangle ABC, the shortest median of this triangle is 10 cm.

4) Three points D, E and F are on sides of BC, BA and AC respectively of a triangle of ABC such that \(BD = DE\) and \(DF = DC\). Find the angle ∠EDF if angle ∠BAC = 40°.

5) Three points D, E and F are on sides of BC, BA and AC respectively of a triangle of ABC such that \(BD = DE\) and \(DF = DC\). Find the angle ∠BAC if angle ∠EDF = 120°.

6) Three points D, E and F are on sides of BC, BA and AC respectively of a triangle of ABC such that \(BD = DE\) and \(DF = DC\). Line BA extends to G then Find the angle ∠GAC if angle ∠EDF = 70°.

7) Two concentric circles having centre O. AB is the diameter of big circle cuts the small circle at D and E. G is a point on bigger circle and F is on smaller circle. Line AG and EF are produced to meet a point C which is outside of bigger circle. Find angle ∠GOF if ∠ACE = 50°.

8) Two concentric circles having centre O. AB is the diameter of big circle cuts the small circle at D and E. G is a point on bigger circle and F is on smaller circle. Line AG and EF are produced to meet a point C which is outside of bigger circle. Find angle ∠ACE if ∠GOF = 90°.

9) In a triangle ABC, D is a point on side BC such that \(BD = DA = AC\). Line BA is extended to E, find angle ∠ADC if angle ∠EAC = 126°.

10) In a triangle ABC, D is a point on side BC such that \(BD = DA = AC\). Line BA is extended to E, find angle ∠EAC if angle ∠ACD = 70°.

11) In a triangle ABC, D is a point on side BC such that \(BD = DA = AC\). Line BA to E, find angle ∠DAC if angle ∠EAC = 120°.

12) In a triangle ABC, points D and E are on side BC and F and G are on side BA such that \(BD = DG = GC = CA = AE = EF = FB\). Find angle ∠GOE if AE and CG intersect at O.

13) In a triangle ABC, two points D and E are on sides BC and BA respectively such that \(BD = DA = AC = EC = EB\). Find angle ∠EOD if AD and CE intersect at O.