Question

Consider a circle with AB as diameter, and P another point on the circle. Let M be the foot of the perpendicular from P to AB. Draw the circles which have AM and MB respectively as diameters, which meet AP at Q and BP at R. Prove that QR is a tangent to both circles.

Answer 1

The figure is given below. Here, O and N are the centers of the smaller circles respectively.

We'll be calling and just for the sake of convenience.

.............................................(1)

In the circle with center O, we have

Similarly, in the circle with center N, we have

Now, in the quadrilateral QONR, we have

So, and are supplementary. But they are the co-interior angles for the line ON.

Therefore, QO||RN.

Now,

( since OQ=OM or is isosceles)

( since OQ=OA or is isosceles)

So,

Similarly, we have

( since NR=NM or is isosceles)

( since NR=NB or is isosceles)

So,

Thus, in thw quadrilateral PQMR, we have

So, PQMR is a rectangle. This implie that PQ||RM and PR||QM.

In we have

.............................. (2)

In the rectangle PQMR, PM and QR are diagonals. So, .

We know that . So,

(using equation 2)

So, we have

This shows that QR is tangent to the circle with center O.

This shows that QR is tangent to the circle with center N.