Question

If C is the mid point of the line segment AB . L and M are the mid point of the line segment AC and BC respectively

Answer 1

Let AB = t units.

Let L, N be the mid points of AM,MB respectively.

Let C (0, ) be the given circle which touches the semi circles with centres L,

M, N at P, R and Q respectively.

Now join OL and ON.

The points O, P, L are collinear and the points O, Q,  N are collinear.

By similarity points M, O, R are collinear

Here   AL = LM = MN = NB = t/4 units.

[since M is the mid point of AB, L is the mid point of AM and N is the mid point of MB]

          Also OL = OP + LP

                     = r + t/4                                [since LP = t/4]

And    ON = OQ + QN

          ON = r + t/4

Þ ΔOLN is an isosceles triangle and M is the mid point of the base LN.

Þ OM ^ LN

\ OMN is a right angled triangle.

Thanks