Name: Areas of Similar Triangles
Uploaded: 2019-07-19
Duration: 6 min 54 s
Description: The corresponding sides of similar triangles are proportional. Are the ratios of areas of similar triangles the same as that of their sides? Watch this video...

Question

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at right angles in $E$ . Prove that $E A^{2} + E B^{2} + E C^{2} + E D^{2} = 4 R^{2},$ where $R$ is the radius of the circumscribing circle.

Accepted Answer

Let $O$ be the centre of the circle and $B D$ the feet of perpendiculars from $O$ to $A C$ and $B D$ respectively.

Clearly $O P E Q$ is a rectangle.

Now, $E A^{2} + E C^{2}$

$= {(E P + P A)}^{2} + {(P C - P E)}^{2}$

$= E P^{2} + P A^{2} + 2 E P \cdot P A + P C^{2} + P E^{2} - 2 P C \cdot P E$

$= 2 (P A^{2} + P E^{2})$ because $P A = P C$ as perpendicular from centre of circle to any chord of it bisects it.

Similarly, $E B^{2} + E D^{2} = 2 (Q D^{2} + Q E^{2})$ .

$∴ E A^{2} + E B^{2} + E C^{2} + E D^{2}$

$= 2 (P A^{2} + P E^{2}) + 2 (Q D^{2} + Q E^{2})$

$= 2 (P A^{2} + O Q^{2}) + 2 (Q D^{2} + O P^{2})$ (since $O P E Q$ is a rectangle so opposite sides are equal)

$= 2 \{(P A^{2} + O P^{2}) + (Q D^{2} + O Q^{2})\}$

$= 2 (O A^{2} + O D^{2})$ (by pythagoras theorem)

$= 2 (R^{2} + R^{2})$

$= 4 R^{2}$

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at right angles in $E$ . Prove that $E A^{2} + E B^{2} + E C^{2} + E D^{2} = 4 R^{2},$ where $R$ is the radius of the circumscribing circle.

Important Questions on Geometry

Learn and Prepare for any exam you want !

The diagonals AC and BD of a cyclic quadrilateral ABCD meet at right angles in E. Prove that EA2+EB2+EC2+ED2=4R2, where R is the radius of the circumscribing circle.

Important Questions on Geometry

Learn and Prepare for any exam you want !

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at right angles in $E$ . Prove that $E A^{2} + E B^{2} + E C^{2} + E D^{2} = 4 R^{2},$ where $R$ is the radius of the circumscribing circle.