The orthocenter of a triangle is the intersection of the three altitudes , that is, the intersection of the lines from each vertex of the triangle to its opposite side forming a right angle. The length of the altitude is the distance between the vertex and the base.
Orthocenter of a triangle
The three altitudes of a triangle meet at a point. That point is called the orthocenter of the triangle .
Specifically: In the drawing are the heights, the orthocenter of the triangle.
To determine the orthocenter of a triangle, we find the intersection of the two altitudes in that triangle.
Note: a) If the triangle is an acute triangle, the orthocenter lies inside the triangle.
b) If the triangle is a right triangle at then the orthocenter coincides with the point .
c) If a triangle is an obtuse triangle, then the orthocenter lies outside the triangle.
Property 1: In an equilateral triangle, the centroid, orthocenter, a point equidistant from the three vertices of the triangle, a point inside the triangle and equidistant from the three sides of the triangle are four coincident points.
Property 2: The orthocenter cuts the perpendicular bisector of two sides into two segments of equal length. This means that the orthocenter is equal distance from the vertices of the triangle.
Property 3: The orthocenter is the center of the circumcircle of a triangle, meaning that if we draw a circle passing through the three vertices of a triangle, the orthocenter will be the center of that circle.
Property 4: The orthocenter of an acute triangle lies inside the triangle, while the orthocenter of an obtuse triangle lies outside the triangle.
Property 5: The orthocenter of a right triangle coincides with the vertex of the right angle of that right triangle.
Property 6: The orthocenter is the only point in a triangle such that if we draw lines from the orthocenter to the vertices of the triangle, the sum of the lengths of those lines is the smallest. This means that the orthocenter is closest to the vertices of the triangle than any other point.
Property 7: The orthocenter is also the center of the circumcircle of the triangle, that is, the largest circle that can be drawn through the three vertices of the triangle.
For example: Given a non-square triangle. Call it its orthocenter. Point out the altitudes of the triangle. From there, point out the orthocenter of the triangle.
Solution guide
Illustration
Let be the feet of the perpendiculars drawn from ΔABC.
Consider ΔHBC with:
so AD is the height from H to BC.
at F so BA is the altitude from B to HC
at E so CA is the height from C to HB.
intersect at A so A is the orthocenter of ΔHCB.
For example: Given a right triangle at with height . Let the midpoint of be , the midpoint of is . Determine the orthocenter of the triangle .
Solution guide
Consider the sub-problem if the triangle has and AC as the midpoints respectively then and .
Indeed, on the opposite ray of the ray take a point such that
Consider triangle AMN and triangle CPN.
(opposite)
, (two sides and two corresponding angles)
Two angles are in alternate positions so
=>(two alternate interior angles)
Consider triangle BMC and triangle PCM.
(cmt)
MC is a common edge
, (corresponding sides and angles)
Two angles are in alternate positions so
We have again
Consider triangle HAB with:
(as proven above)
Consider triangle ADE.
on the other hand and
is the height of triangle ADE
C is the intersection of AC and DC
=> C is the orthocenter of triangle ADE
For example: Given a triangle at A, the altitude intersects the median at . Prove and calculate?
Instruct
Illustration
Because the balance is at A and AM is the median
⇒ AM is also the altitude corresponding to BC
at M.
On the other hand, and so K is the orthocenter.
Therefore, K belongs to the altitude from C of ∆ABC.
We have:
Plan an inclusive virtual Halloween party for remote teams with time-zone-friendly scheduling, simple games, accessible options, and a practical hosting checklist.
Explore seven Halloween team building games for the workplace, with quick rules, hybrid-friendly options, accessibility tips, and practical safety guidance.
Mix a glossy prop-only fake blood batch, choose safer products for skin and costumes, and test cleanup before Halloween night.
Create a gentle Halloween look with pastel colors, friendly ghosts, easy no-carve accents, and practical safety tips for your home.
Create a spooky office cubicle with Halloween desk decor ideas that keep your workspace usable, tidy, and mindful of workplace safety.
Plan a recognizable Halloween 2026 look with pop culture costumes inspired by Spider-Man, Supergirl, Toy Story 5, Moana, Mario, and Wednesday.
Sort leftover Halloween candy safely, share or repurpose sealed treats, and choose food use or compost for pumpkins based on their condition.
Compare workplace Halloween costume choices by respect, dress code, mobility, and safety—and learn when a subtle accessory or a policy check is the better option.
Find a couples costume that fits your style, budget, and comfort level, from Addams Family classics to original inside-joke duos.
Need a last-minute high school Halloween costume? Compare 10 easy closet-based ideas by prep time, comfort, school-day practicality, and simple safety checks.
