Question

triangle abc is an equilateral triangle angle b is bisector of angle a prove that cq =ca

Answer 1

AB = AC

ACB =ABC ..........1

ACB is exterior angle of triangle AQC

ACB=QAC+AQC

ABC=QAC+AQC ........from 1 ....2

Now ABP=PBC.....angle B is bisector

2PBC=ABC

hence, 2 PBC = QAC + AQC.......from 2.....3

PAC = PBC ...... tangent secant theorem.......4

hence, PBC = AQC ..........5

2 PBC = PBC + AQC ....... from 3 and 5

PBC = AQC

PAC = AQC ......from 4

QAC = AQC

Therefore, CQ = CA ..... side opposite to equal sides are equal.

HENCE PROVED