- #1
Albert1
- 1,221
- 0
Point $H$ is the orthocenter of $\triangle ABC$
prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$
prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$
Proving Orthocenter Property of Triangle ABC
In summary, the conversation discusses the proof that in a triangle $ABC$, with orthocenter $H$, the sum of the squares of the distances from $H$ to the vertices equals the sum of the squares of the lengths of the sides. The solution involves using the fact that $PAHB$ is a parallelogram and $PB$ is perpendicular to $BC$, resulting in the equation $HA^2+BC^2=4R^2$, where $R$ is the radius of the circumcircle. It is stated that this solution is obvious and the conversation ends with confirmation that the problem has been solved.
- #2
- #3
caffeinemachine
Gold Member
MHB
- 816
- 15
Very nice solution.Albert said:
It is clear that $PAHB$ is a parallelogram and that $PB$ is perpendicular to $BC$.
Thus $HA^2+BC^2=4R^2$, where $R$ is the radius of the circumcircle.
- #4
Albert1
- 1,221
- 0
yes, you got it !caffeinemachine said:
Related to Proving Orthocenter Property of Triangle ABC
1. What is the orthocenter property of a triangle?
The orthocenter property of a triangle states that the three altitudes of a triangle intersect at a single point, known as the orthocenter.
2. How do you prove the orthocenter property of a triangle?
The orthocenter property of a triangle can be proved using several methods, such as coordinate geometry, vector algebra, or trigonometry. One common method is to use the perpendicular bisectors of the sides of the triangle to show that they intersect at a single point, which is the orthocenter.
3. Can you prove the orthocenter property of a triangle using only geometry?
Yes, the orthocenter property of a triangle can be proved using only geometry. One method is to construct the altitudes of the triangle and show that they intersect at a single point. Another method is to use the properties of perpendicular bisectors to prove the orthocenter property.
4. Is the orthocenter property of a triangle always true?
Yes, the orthocenter property of a triangle is always true. It is a fundamental property of triangles and is not dependent on the shape or size of the triangle.
5. Why is the orthocenter property of a triangle important?
The orthocenter property of a triangle is important in geometry as it helps to define and identify different types of triangles. It also has practical applications in fields such as architecture and engineering, where the orthocenter is used to determine the height of a building or the placement of supports for bridges.
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