Ortho-centre Along with relation between all the centres

Hello maths bugs(🐞) and hivers(🐝)
I hope you are strong and stout.

Today's topic is Ortho-Centre of a triangle any of its kind.Today we gonna finish all the centres of a triangle. People get confused to differentiate them and their properties.In my previous post, I have shared lucid explanation of cir-cumcircle , in-circle , centroid. Please check them first because today I'm going to share relation between all the centres also and if you lack idea of them , reading this post may be useless for you.

What is called an Ortho-centre?

A point where meet all the three perpendiculars from the vertices to its opposite sides or extended opposite sides is called an Ortho-centre. Check it in the figure below.

img_0.6947944511141309

Points to be considered:

  1. The Ortho-centre may be inside or inside or on the side of a triangle.
  2. There is no circle can be drawn whose centre is the Ortho-centre and the circle passes through the vertices unless it is not a equilateral triangle.
  3. All the perpendiculars drawn from vertices to their opposite sides may not be the bisectors of respective sides unless it is an equilateral βˆ†.

βœ…βœ… Ortho-centre of a right angle triangle is shown below:(Yes, it is at angle 90Β°)
img_0.4534987634049827

βœ…βœ…Ortho-centre of a obtuse angle triangle is as below:(Yes, it is outside the βˆ†)
img_0.10595728121687903
βœ…βœ…Ortho-centre is inside of a acute angle triangle as shown in the figure below:

img_0.2705212568729492

RELATION BETWEEN CENTRES OF A βˆ†:

β˜‘οΈβ˜‘οΈ All the centres come at the same point inside of a equilateral βˆ†.At that time distance between them is zero.

β˜‘οΈβ˜‘οΈ C G O i.e cir-cum-centre , geo-centre/centroid and Ortho-centre come in a straight line irrespective of which kind of triangle it is(an equilateral βˆ†). In- centre may not come in the same straight line

β˜‘οΈβ˜‘οΈ Ratio of distance between cir-cum circle & centroid and centroid & Ortho-centre is 2:1.

β˜‘οΈβ˜‘οΈ Distance between in-centre and cir-cum-centre of a triangle(except an equilateral βˆ†) is square root of difference of in-radius and cir-cum-radius.Check it in the figure below:

img_0.7205684359314418

All the figures in the post are made by me with math editor

I hope you find my post interesting.

Thank you very much for stopping by.

Next post will be soon.

All is well

Regards: @meta007

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