Baudhayan Theorem:

The Baudhayan theorem is a geometric theorem named after the ancient Indian mathematician Baudhayan. It is a special case of the Pythagorean theorem and provides a formula to calculate the area of an isosceles trapezoid.

Statement of the Theorem:

In an isosceles trapezoid, the sum of the squares of the diagonals is equal to the square of the base added to four times the square of the distance between the midpoints of the parallel sides.

Proof of the Theorem:

Step 1: Consider an isosceles trapezoid ABCD where AB and CD are parallel sides, AD and BC are the non-parallel sides, and AC and BD are the diagonals. Let E and F be the midpoints of AD and BC, respectively.

Step 2: Draw perpendiculars from E and F to AB and CD, respectively. Let the points of intersection be G and H.

Step 3: Since E and F are midpoints, EG = GH = HF = FE.

Step 4: Let AG = a, BG = b, CH = c, and DH = d.

Step 5: Using the Pythagorean theorem, we can find the lengths of the diagonals:

AC² = a² + EG² (1)
BD² = b² + EG² (2)

Step 6: Now, consider the triangles AEG and CFH. They are congruent by SAS (side-angle-side) congruence.

Step 7: Therefore, AF = EG = GH = HF = FE.

Step 8: Using the Pythagorean theorem, we can find the lengths of the diagonals:

AC² = AF² + CF² (3)
BD² = BF² + CD² (4)

Step 9: Since AB is parallel to CD, triangles AGB and CHD are similar.

Step 10: Using the property of similar triangles, we can write:

AG/CH = BG/DH

Step 11: Squaring both sides, we get:

(AG/CH)² = (BG/DH)²

Step 12: Substituting the values from Step 4, we have:

(a/c)² = (b/d)²

Step 13: Cross-multiplying, we get:

a²d² = b²c²

Step 14: Adding equations (1), (2), (3), and (4), we obtain:

AC² + BD² = 2(EG² + AF² + BF² + CF²)

Step 15: Substituting the values from Step 5 and Step 8, we have:

AC² + BD² = 2(a² + EG² + b² + EG²)

Step 16: Simplifying, we get:

AC² + BD² = 2(a² + b² + 2EG²)

Step 17: Using the property from

It's Pythagoras theorem dude