Q35 of 62 Page 10

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° – ഊ, 90° – ഋ and 90°–ഌ.

Given: Bisectors of angles A, B and C of a triangle ABC intersect its circum circle at D, E and F respectively.
To Prove: The angles of the ΔDEF are   90° –  ,   90° – and 90° –   respectively.
                                                                                                             
Construction: Join DE, EF and FD.
Proof: ∠ FDE = ∠ FDA + ∠ EDA
                   = ∠ FCA + ∠ EBA | Since angles in the same segment are equal
                   = ∠ C + ∠ B
                   =∠ D = =       | In Δ ABC, ∠ A + ∠ B + ∠ C = 180°  (Angle Sum Property)
                   = 90° – 
Similarly, we can show that
∠ E =  90° –   
and ∠ F = 90° – 

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33 ABCD is a parallelogram. The circle through A, B, and C intersect CD (produced if necessary) at E. Prove that AE = AD. 34 AC and BD are chords of a circle which bisect each other. Prove that AC and BD are diameters and ABCD is a rectangle. 36 Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ. 37 In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.