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Suppose ΔABC is triangle right angled at B.

Let line-segment BD⊥AC

When an altitude is drawn to the hypotenuse of a right angled triangle, the length of altitude is the Geometric Mean of segments of hypotenuse made by altitude on hypotenuse and at the same time each remaining side of triangle is Geometric Mean of hypotenuse and the segment of Hypotenuse adjacent to it.

∴AB²= AD*AC

And BC²= CD*AC

So, AB²+BC²= AD*AC + CD*AC

= AC(AD+CD)

= AC*AC

So, AB²+BC²=AC²

Let line-segment BD⊥AC

When an altitude is drawn to the hypotenuse of a right angled triangle, the length of altitude is the Geometric Mean of segments of hypotenuse made by altitude on hypotenuse and at the same time each remaining side of triangle is Geometric Mean of hypotenuse and the segment of Hypotenuse adjacent to it.

∴AB²= AD*AC

And BC²= CD*AC

So, AB²+BC²= AD*AC + CD*AC

= AC(AD+CD)

= AC*AC

So, AB²+BC²=AC²