[royprosen]

When I asked for the proof of the theorem that among all the triangles of equal base and area,the perimeter of the isosceles triangle is the least,you proved it using calculus.But for a student of tenth standard,it is beyond reach.Please give the proof using only the theorems of pure geometry.

[ebaines]

Here's one way - not sure it's purely a geometry approach, as it involves a little knowledge of conic sections.

Consider the base line B, and imagine that you have a string of fixed length with the ends of the string attached to the ends of the baseline B. If you press a pencil against the string and stretch it as far as you can, and then draw the locus of points that results, you have an ellipse, whose semi-minor axis is centered above the midpoint of the baseline. The semi-minor axis is the part of an ellipse which is furthest in vertical distance from the baseline. Hence the area of the triangle defined by the baseline and that point on the ellipse is the largest possible area for that patricular string length. Hence for a triangle of fixed perimeter, the largest triangle area is the isosceles one. Or conversely, for a triangle of fixed area, the shortest possible perimeter is an isosceles triangle. Hope this helps.

[galactus]

Have you considered trying Heron's formula? That is the way I have seen these done before.