*5 Orthocenter Things That Geometry Kids Should Know

Directions

Drag the vertices of the triangle and see how each fact holds true. Then answer the two questions at the bottom of the page.

FACT 1: The orthocenter cuts each altitude into two parts such that the products of each part of the altitude is equal to the product of the parts of the other altitudes.

[size=200]AH x HF = CH x HD = BH x HE[/size]

FACT 2: Distance from orthocenter to vertex, length of opposite side, & the circumcircle's diameter are all related.

[size=200]AH[sup]2[/sup] + BC[sup]2[/sup] = BH[sup]2 [/sup]+ AC[sup]2[/sup] = CH[sup]2 [/sup]+ AB[sup]2[/sup] = Diameter[sup]2[/sup] [br][/size]

FACT 3: Distances from the orthocenter to the vertices are related to the diameters of the incircle and circumcircle.

[size=200]AH + BH + CH = Incircle Diameter + Circumcircle Diameter[/size]

FACT 4: Distances from the vertices to the orthocenter is linked to the distance from the circumcenter to the opposite side.

[size=200]Distance from orthocenter (H) to vertex = 2 x Distance from circumcenter (D) to opposite side[/size]

FACT 5: Orthocentric Quartet

[size=200]Given 3 points and their orthocenter, any of the 4 points is the orthocenter of the other 3 points.[/size]

Consider the 5 Facts Above

[color=#bf9000]Select one of the facts and explain it in a way such that a middle schooler could understand it. (Assume that the middle-schooler has a solid grasp of Pythagorean Theorem). [/color]

Still thinking about the 5 facts above...

Which fact do you find trickiest to understand?

FACT 2: How the Distance from Orthocenter to Vertex, Length of Opposite Side, Circumcircle Diameter relate to each other. FACT 1: The Orthocenter cuts each altitude into two parts whose products for each altitude are equal. FACT 4: Distance from orthocenter to vertex = 2x distance from circumcenter to opposite side. FACT 5: Orthocentric Quartet FACT 3: Distances from the Orthocenter to the Vertices.