What is the orthic triangle of ABC?

What is the orthic triangle of ABC?

The orthic triangleof ABC is defined to be A*B*C*. Thistriangle has some remarkable properties that we shall prove: The altitudes and sides of ABC are interior and exterior angle bisectors of orthic triangle A*B*C*, so H is the incenter of A*B*C* and A, B, C are the 3 ecenters (centers of escribed circles).

What are the excenters of triangle ABC?

H, I, O, N are orthogonal center, incenter, circumcenter, and Nagelian point of triangle ABC. Ia , Ib , Ic are excenters of ABC corresponding vertices A, B, C. S is point that O is midpoint of HS.

How do you prove triangle ABC isosceles?

Triangle ABC is isosceles (AB = AC). From A, we draw a line ` parallel to BC. P, Q are on perpendicular bisectors of AB, AC such that P Q ⊥ BC. π M, N are points on ` such that angles ∠AP M and ∠AQN are . Prove that 2 1 1 2 + ≤ AM AN AB 15. In triangle ABC, M is midpoint of AC, and D is a point on BC such that DB = DM .

How do you prove that circum circle of ILK is tangent to ABC?

Incircle of triangle ABC touches AB, AC at P, Q. BI, CI intersect with P Q at K, L. Prove that circumcircle of ILK is tangent to incircle of ABC if and only if AB + AC = 3BC. 2 f 11. Let M and N be two points inside triangle ABC such that ∠M AB = ∠N AC and ∠M BA = ∠N BC. Prove that AM · AN BM · BN CM · CN + + = 1.

Which orthic triangle has the smallest perimeter?

The sides of the orthic triangle form an “optical” or “billiard” pathreflecting off the sides of ABC. From this it can be proved that the orthic triangle A*B*C* has the smallest perimeterof any triangle with vertices on the sides of ABC.

How do you prove that all three angles are congruent?

Angle ACC** is an inscribed angle subtending the same arc asangle ABC**, so these two angles are equal. Thus all 3 angles are congruent: angle C*BH = angle ACC* = angle C*BC**. Applying this proposition to each altitude, we get thistheorem.

How do you find the altitude of a triangle with acute angles?

Altitudes and the Orthic Triangle of Triangle ABC Altitudes and the Orthic Triangle of Triangle ABC Given a triangle ABC with acute angles, let A*, B*, C* bethe feet of the altitudes of the triangle: A*, B*, C* are points on the sidesof the triangle so that AA* BB*, CC* are altitudes.