P, Q are points on the side AC, AB of triangle ABC such that
angBPC = ang BQC; BP cuts CQ at K; X, Y are points on AC, AB such that KXAY is a parallelogram. Prove that (i) AX.XC = AY.YB (ii) the centre of the circle ABC is equidistant from X and Y
I hav already solve (i) but just to typed if it is of any help to (ii)
PS sorry but i dont know how to draw diagrams on computer
since angBQC = angCPB (given) and are on the same side of BC, the pts BQPC are concylic (equal angles subtended on same side of a line)
Therefore, angYBK=angQBK = ang XCK=angXCK (angles subtended on same arc QP)
In parallelogram AYKX, angAYK = angAXK (opp ang of parallelgram)
Therefore angBYK = 180 deg - angAYK = 180 deg - angAXK = angCXK
Then it can be proven with angBYK = angCXK and angYBK = angXCK that the triangles YKB and XKC are equiangular
thus YK / XK = YB / XC (sides proportional in eqiangular tri)
YK . XC = YB . XK
But YK = AX (opp sides of parallelogram) and similiarly AY= XK
AX. XC = AY. YB
Hope that helps to do part (ii) because i really hav no idea
I've attached a sketch of this problem.
1. The angles and are equal and they are subtended over the same side of the triangle thus the points B, Q, P, C must lie on a circle with as centre (green).
2. Therefore you can use the intersecting chord theorem here, using BP and CQ as chords and K as intersection.
3. The triangles YQK and PXK are similar.
4. With the green circle and the secants AC and AB you can use the secant-secant-theorem.
5. The two centres and must lie on a perpendicular bisector of BC. And must lie on a perpendicular bisector of XY.
But: I haven't got a clue how to put all those facts together to proof the statement of your problem.
regarding to oreilly, it was a typing mistake that i made.
earthboth, i dont get what that intersecting chord theorem in 2 is used for, also i dont get what the secant-secant theorm is and used for either,
however, it will be nice if u can provide the proof that
must lie on a perpendicular bisector of XY as this is the prove that the centre is eqidistant from XY
Thank you so much for both ur help