In a non right triangle P Q R , the median from R meets the sides P Q at S , the

Question

In a non right triangle

P Q R

, the median from

R

meets the sides

P Q

at

S

, the perpendicular from

P

meets sides

Q R

at

E

and

R S

and

P E

intersect at

O

.

p = \sqrt{3}, q = 1

and circumradius of

P Q R

is

1

.

Answer 1

Note that

P E

is a median and hence

P E = 3 O E

. So

△ P Q R = 3 \times △ O Q R

.
The question states that

△ P Q R

is not a right angled triangle otherwise

∠ R = 90^{0}

is another obvious configuration for the given sides and circumradius.
As

p = \sqrt{3} > q = 1, ∠ Q

is acute. That leads to only other possible configuration of

∠ P

being obtuse.
If

C

is the circumcenter, you can see why

∠ P = 120^{0} P Q = P R

and

E

is the midpoint of

Q R

.

Answers (1)