show that the locus of a point which moves in such a way that square of its distance from the base of an isosceles triangle is equal to the rectangle under its distance from other sides is a circle.

I can do this by coordinate geometry, but I don't see a synthetic (euclidean) proof of it.

If the triangle is PQR, with PQ = PR, take P, Q, R to be the points (0,b), (–a,0), (a,0) respectively. The equation of the line PQ is \(\displaystyle bx+ay-ab=0\) and the equation of PR is \(\displaystyle bx -ay +ab = 0\).

If I understand the question correctly, you want to find the locus of a point X = (x,y) which moves so that the square of its distance from QR is equal to the product of its distances from PQ and PR.

With the above choices of P, Q and R, the distance from X to PQ is \(\displaystyle \frac{a(b-y)+bx}{\sqrt{a^2+b^2}}\), and the distance from X to PR is \(\displaystyle \frac{a(b-y)-bx}{\sqrt{a^2+b^2}}\). The distance from X to QR is |y|. So the locus of X is given by the equation \(\displaystyle \dfrac{a^2(b-y)^2 - b^2x^2}{a^2+b^2} = y^2\). This simplifies to \(\displaystyle x^2 + \bigl(y+\frac{a^2}b\bigr)^2 = \frac{a^2}{b^2}(a^2+b^2)\), which represents a circle with centre at \(\displaystyle \bigl(0,-\frac{a^2}b\bigr)\) and radius \(\displaystyle \frac ab\sqrt{a^2+b^2}\).

In terms of the geometry of the picture, this circle has its centre C on the perpendicular bisector of QR. It passes through Q and R, and is tangential to PQ and PR.