Question

If in an isosceles right angled triangle the area of circumcircle is 16 times the area of incircle, also all points circumcenter, orthocenter, incenter, centroid lie in the 1st quadrant, orthocenter being at origin. Find the distance between circumcenter and incenter.

• A. 4R - 2 $\sqrt{2}$ r
• B. 4r - 2 $\sqrt{2}$ r
• C. 4R - 2 $\sqrt{2}$ R
• D. 4 - 2 $\sqrt{2}$ r

Answer 1

The correct option is B 4r - 2 $\sqrt{2}$ r

It is given that the area of circumcircle is 16 times of that of incircle.
⇒ radius of circumcircle is 4 times the radius of incircle.
or R = 4r

Distance between Orthocenter and Circumcenter = R $\sqrt{1-8cosA.cosB.cosC}$
The triangle being an right angled triangle distance OC = R

We know, that distance between Orthocenter and incenter = $\sqrt{R^2-2Rr}$
R = 4r
$=\sqrt{16r^2-8r^2}$
$=2\sqrt{2}r$

It is given that the triangle is isosceles. And we know that, in isosceles triangle all the points (O,G,C,I) lie on the same line.
Since all the points are in 1st quadrant and orthocenter is at origin,Distance between CI = OC - OI
OI = 2 $\sqrt{2}r$
OC = 4r

So, CI = 4r - 2 $\sqrt{2}r$