Triangle Orthocenter Calculator

Enter the three points of the triangle：
   X   Y
A:
B:
C:
 
Orthocenter of triangle：

Let the three altitudes of ⊿ABC be AD, BE, and CF, among which D, E, and F are the feet of the perpendicular, the orthocenter is H, and the opposite sides of angles A, B, and C are a, b, and c respectively

1. The orthocenter of an acute triangle is inside the triangle; the orthocenter of a right triangle is at the vertex of the right angle; the orthocenter of an obtuse triangle is outside the triangle.

2. The orthocenter of a triangle is the incenter of its foot triangle; or, the incenter of a triangle is the orthocenter of its circumcenter triangle; 3. The symmetric points of the orthocenter H about the three sides are all on the circumscribed circle of △ABC.

4. In △ABC, there are six groups of four points on the same circle, three groups (four in each group) of similar right triangles, and AH·HD=BH·HE=CH·HF.

5. Any point among H, A, B, C is the orthocenter of the triangle with the other three points as vertices (and such four points are called an orthocenter group).

6. The circumscribed circles of △ABC, △ABH, △BCH, and △ACH are equal circles.

7. In a non-right triangle, the straight line through H intersects the straight lines AB and AC at P and Q respectively, then AB/AP·tanB+AC/AQ·tanC=tanA+tanB+tanC.

8. The distance from any vertex of a triangle to the orthocenter is equal to twice the distance from the circumcenter to the opposite side.

9. Let O and H be the circumcenter and orthocenter of △ABC respectively, then ∠BAO=∠HAC, ∠ABH=∠OBC, ∠BCO=∠HCA.

10. The sum of the distances from the orthocenter of an acute triangle to the three vertices is equal to twice the sum of the radii of its incircle and circumcircle.

11. The orthocenter of an acute triangle is the incenter of the foot triangle; among the inscribed triangles of an acute triangle (vertices on the sides of the original triangle), the foot triangle has the shortest perimeter.

12. Simson's Theorem (Simson Line): The necessary and sufficient condition for the feet of the perpendicular lines drawn from a point to the three sides of a triangle to be collinear is that the point falls on the circumcircle of the triangle.

13. Suppose there is a point P inside the acute angle ⊿ABC, then the necessary and sufficient condition for P to be the orthocenter is PB*PC*BC+PB*PA*AB+PA*PC*AC=AB*BC*CA.