Thread: Need help on Circle properties

In the diagram, the circle ABC passes through the centre X of circle ABD, and CBD is a straight line. Prove that triangle ACD is isosceles.

Hello, acc100jt!

I think I've got it . . .

In the diagram, the circle ABC passes through the centre X of circle ABD,
and CBD is a straight line.
Prove that triangle ACD is isosceles.

Draw radii XA, XB, and XD.

∆XAD is isosceles: ./XAD = /XDA = α

∆XBD is isoscles: ./XDB = /XBD = β . → . /CBX .= .180° - B

Hence: ./D .= ./XDA + /XDB .= .α + β


CAXB is a cyclic quadrilateral; its opposite angles are supplementary.
Since /CBX = 180° - β, then /CAX = β.

Hence: ./A .= ./XAD + /XAC .= .α + β


Therefore: ./A = /D, .CA = CD . → . ∆ACD is isosceles.