Different Circles and Centers Connected with Triangle – Orthocenter

Different Circles and Centers Connected with Triangle – Orthocenter

Orthocenter(H) is the point of intersection of the altitudes of a triangle.

Orthocenter (H) of an acute angled triangle lies inside the triangle.

Orthocenter (H) of an obtuse angled triangle lies inside the triangle.

Orthocenter(H) of a right-angled triangle ABC lies at the right angle itself. In fig, the orthocenter H coincide with the right-angle B.

Image if orthocenter(H) in any side of a triangle lies on the circumcircle.

$$\angle HBD=\angle EBC=\left( \frac{\pi }{2} \right)-C$$,

$$\Rightarrow \angle BHD=C$$,

$$Also,\ \ \angle BPD=\angle BPA=\angle BCA=C$$,

$$Thus,\ \ \Delta BPD\ \ and\ \ \Delta BHD\ \ are\ \ congruent$$,

⇒ HD = DP

⇒ P is image in H in BC.