Hii

Construction = AE is perpendicular BC

PROOF

In ∆ABC

AB^2 = AE^2 +BE^2

OR

AB^2 = AD^2 - DE^ 2 + (BD - DE) ^2

= AD^2 - DE^2 + BD^ 2 + DE^2 - 2BD * DE

= AB^2 + BD^2 - 2BD*DE.....1

IN ∆ AEC

AC^2 = AE^2 + EC^2

OR

AC^2 = AD^2 + DC^2 + 2ED*DC......2

ADDING 1 N 2

AB^2 + AC^2 = 2 (AD^2 + BD ^2) (BD= DC)

= 2 AD^2 +2 *(1/2 BC) ^2

= 2AD^2 + 1/2 BC ^2

HENCE PROVED..

Given:
Triangle ABC
AD is a median
AE is perpendicular to BC

To prove:
AB^2 AC^2 = 2AD^2 1/2BC^2

Proof:

1. Drawing the diagram:
Let's start by drawing the given triangle ABC and labeling the points as mentioned.

2. Understanding the properties:
To prove the given equation, we need to apply the properties of the median and the perpendicular bisector.

3. Properties of a median:
A median in a triangle divides the opposite side into two equal segments.

In triangle ABC, AD is a median, so it divides side BC into two equal segments, BD and DC.

4. Properties of a perpendicular bisector:
A perpendicular bisector of a line segment is a line that divides the segment into two equal parts and is perpendicular to it.

In triangle ABC, AE is a perpendicular bisector of side BC. It divides BC into two equal parts, BE and EC, and is perpendicular to BC.

5. Using the properties:
We can now use the properties of the median and the perpendicular bisector to prove the given equation.

Since AD is a median, we have BD = DC.

Since AE is a perpendicular bisector, we have BE = EC.

6. Using the Pythagorean theorem:
We can apply the Pythagorean theorem in triangles ABD and AEC to further simplify the equation.

In triangle ABD, applying the Pythagorean theorem, we have:
AB^2 = AD^2 + BD^2

In triangle AEC, applying the Pythagorean theorem, we have:
AC^2 = AE^2 + EC^2

Since BD = DC and BE = EC, we can rewrite the equations as:
AB^2 = AD^2 + (1/2)BC^2
AC^2 = AE^2 + (1/2)BC^2

7. Substituting the values:
We can substitute the values of AB^2 and AC^2 from the above equations into the given equation.

AB^2 AC^2 = 2AD^2 1/2BC^2

Substituting the values, we get:
(AD^2 + (1/2)BC^2)(AE^2 + (1/2)BC^2) = 2AD^2 1/2BC^2

8. Simplifying the equation:
Expanding the equation, we get:
AD^2AE^2 + (1/2)BC^2AD^2 + (1/2)BC^2AE^2 + (1/4)BC^4 = 2AD^2 1/2BC^2

Now, let's simplify the equation further.

9. Canceling out the common terms:
We can cancel out the common terms on both sides of the equation.

AD^2AE^2 + (1/2)BC^2AD^2 + (1/2)BC^2AE^2 + (1/4)BC^4 = 2AD^2 1/2BC^2

AD^2AE^2 + (1/2)BC^