MHB Proving Orthocenter Property of Triangle ABC

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Property Triangle
Click For Summary
The discussion focuses on proving the orthocenter property of triangle ABC, specifically the equation HA² + BC² = HB² + AC² = HC² + AB². Participants note that the configuration of points forms a parallelogram, with PB being perpendicular to BC. This leads to the conclusion that HA² + BC² equals 4R², where R is the circumradius. The solution is acknowledged as clear and effective. The proof highlights the relationship between the orthocenter and the triangle's circumcircle.
Albert1
Messages
1,221
Reaction score
0
Point $H$ is the orthocenter of $\triangle ABC$

prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$
 
Mathematics news on Phys.org
Albert said:
Point $H$ is the orthocenter of $\triangle ABC$

prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$
hint:
A solutiopn of the diagrm of this problem is given,now it is obvious ,hope someone can solve it
 

Attachments

  • Othocenter.png
    Othocenter.png
    3.1 KB · Views: 105
Albert said:
hint:
A solutiopn of the diagrm of this problem is given,now it is obvious ,hope someone can solve it
Very nice solution.

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.
 
caffeinemachine said:
Very nice solution.

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.
yes, you got it !
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

MHB Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$
  • · Replies 4 ·
Replies
4
Views
1K
MHB What are the angles of an isosceles triangle with a specific ratio?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Proving Equality of Angles in an Acute Triangle
  • · Replies 4 ·
Replies
4
Views
1K
High School A triangle problem: Prove that |AC|+|AB|=|PR|+|PQ| given some constraints
  • · Replies 13 ·
Replies
13
Views
2K
MHB Area of Triangle ABC Given Dimensions
  • · Replies 5 ·
Replies
5
Views
2K
MHB Is the Midpoint of HE the Center of the Inscribed Circle in Triangle HBC?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Can you prove that Triangle $ABC$ is Isosceles?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Proof Quest: Non-Equilateral Triangle Enclosing Point Z
  • · Replies 3 ·
Replies
3
Views
2K
MHB Geom Ch: Prove $AB=x^3$ Given $\triangle ABC$ & $\triangle AEF$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove Triangle ABC is a Right Triangle
  • · Replies 1 ·
Replies
1
Views
1K