Question

Q.1.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° – (½)A, 90° – (½)B and 90° – (½)C.

Q.2.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.

Q.3: Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.
​

Answer 1

Question:- 1

Given :-

• Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively.

To Prove :-

• The angles of the triangle DEF are 90° – (½)A, 90° – (½)B and 90° – (½)C.

Proof :-

• Please find the attachment number 1

Question : - 2

Given :-

• In triangle ABC, the angle bisector of ∠A and perpendicular bisector of BC intersect at D.

To prove :-

• D lies on the circumcircle of the triangle ABC.

Construction :-

• Join OB and OC.

Proof :-

• Please find the attachment number 2

Question :- 3

Given :-

• A rhombus ABCD in which a circle is drawn taking AB as diameter. Diagonal AC and BD intersects at O.

To Prove :-

• Circle drawn on AB as diameter passes through O.

Proof :-

Given that

• ABCD is a rhombus having diagonals AC and BD intersect each other at O.

We know,

• Diagonals of rhombus bisect each other at right angles.

$\rm :\longmapsto\: \: \: \therefore \: \angle \: AOB = 90 \degree \:$

Since

$\rm :\longmapsto\: AB \: is \: diameter\: \: and \: \angle \: AOB = 90 \degree \:$

$\bf\implies \:O \: lies \: on \: circumference \: of \: circle.$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. Angle in same segments arw equal.

2. Angle in semi-circle is 90°.

3. Angle subtended at the centre of a circle by an arc is double the angle subtended by the same arc on the circumference.

4. Perpendicular bisector of a chord of a circle passes through centre.

5. Equal chords are equidistant from centres.

6. Equal chords subtends equal angles at centre.