Centres of a Triangle ത്രികോണകേന്ദ്രങ്ങൾ
A triangle has three altitudes and three medians. Draw any one of them and nothing special happens; draw all three and something surprising does: they always pass through a single point. The parallel-line facts of this chapter – especially the midpoint theorem – are exactly what we need to prove it.
- The three perpendicular bisectors of the sides of a triangle meet at one point, the circumcentre (പരിവൃത്തകേന്ദ്രം) – Class 8.
- Opposite sides of a parallelogram are equal; its diagonals bisect each other; a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.
- Midpoint theorem: the line joining the midpoints of two sides of a triangle is parallel to the third side and half of it.
1. Altitudes ഉന്നതികൾ
The perpendicular from a vertex to the opposite side is an altitude (ഉന്നതി, also called the height). Every triangle has three. Drag the corners below and watch where the three altitudes go.
Orange dots can be dragged. For an acute triangle the meeting point is inside; make one angle 90° and it sits on that vertex; make an angle bigger than 90° and it jumps outside (the altitudes are then drawn on the extended sides).
2. Why the three altitudes meet at one point
The textbook's proof is clever: it turns the altitudes of our triangle into the perpendicular bisectors of a bigger triangle, and those we already know meet at one point.
ഏതു ത്രികോണത്തിന്റെയും ഉന്നതികളെല്ലാം ഒറ്റ ബിന്ദുവിലൂടെ കടന്നുപോകും.
The three altitudes of any triangle pass through one point. This point is called the orthocentre (ലംബകേന്ദ്രം).
3. Medians നടുവരകൾ
The line from a vertex to the midpoint of the opposite side is a median (നടുവര). Two medians obviously cross somewhere. Does the third one pass through the same point? The midpoint theorem answers it – and tells us exactly where the point is on each median.
ഏതു ത്രികോണത്തിന്റെയും നടുവരകൾ മൂന്നും ഒറ്റ ബിന്ദുവിലൂടെ കടന്നുപോകും; ഓരോ മൂലയിൽ നിന്നും ഈ ബിന്ദുവിലേക്കുള്ള അകലം, എതിർവശത്തിന്റെ മധ്യബിന്ദുവിൽ നിന്നുള്ള അകലത്തിന്റെ രണ്ടു മടങ്ങാണ്; അതായത്, ഈ ബിന്ദു നടുവരകളെയെല്ലാം 2 : 1 എന്ന അംശബന്ധത്തിൽ ഭാഗിക്കും.
The three medians of any triangle pass through one point; its distance from each vertex is twice its distance from the midpoint of the opposite side – it divides every median in the ratio 2 : 1. This point is the centroid (മധ്യമകേന്ദ്രം).
Lengths are shown in the units of the drawing. Whatever the shape, AG is twice GD, BG is twice GE and CG is twice GF.
Divide the median into 3 equal parts: the centroid is 2 parts from the vertex and 1 part from the midpoint of the opposite side.
4. The Euler line ഓയ്ലർ വര
Three centres have now appeared: the circumcentre O (where the perpendicular bisectors meet), the centroid G and the orthocentre H. The Swiss mathematician Leonhard Euler discovered in the 18th century that they always lie on one straight line, and that G divides OH in the ratio 1 : 2. Test it by dragging.
Blue O = circumcentre (centre of the circle through the three corners), green G = centroid, red H = orthocentre. The purple line through them is the Euler line. Make the triangle equilateral and all three points melt into one.
5. The balance point സമതുലനം
Cut a triangle out of cardboard and try to balance it on the tip of a pencil. There is exactly one spot where it stays level: the centroid. Physicists call it the centre of mass (ദ്രവ്യകേന്ദ്രം). Slide the pencil along a median and see.
The triangle tips towards the side that has more cardboard. Only when the pencil is at the centroid – two-thirds of the way from the vertex – is the cardboard shared equally on every side.
Mark the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from the orthocentre to the three vertices. All nine points lie on one circle! Its centre is the midpoint of OH on the Euler line. Try it in GeoGebra with the "Circle through 3 Points" tool.
A waiter balances a triangular tray by holding it under its centroid; a kite maker ties the string so that the weight is spread evenly about the centre of mass; engineers compute the centroid of every beam cross-section before deciding how it will bend. Roof trusses and bridge panels are triangles, and their centroids tell the designer where the load acts. Even a flat shape as odd as the map of India has a balance point – and it is found by cutting the shape into triangles.
6. Check yourself
- Altitude = perpendicular from a vertex to the opposite side. The three altitudes meet at the orthocentre: inside an acute triangle, at the right-angle vertex of a right triangle, outside an obtuse triangle.
- Proof: parallels through the vertices make a bigger triangle whose perpendicular bisectors are our altitudes.
- Median = vertex to midpoint of the opposite side. The three medians meet at the centroid, which divides each median 2 : 1 (vertex side is the longer piece).
- Proof: with P, Q the midpoints of AG, BG, the quadrilateral PQDE is a parallelogram (midpoint theorem twice).
- Circumcentre, centroid and orthocentre lie on the Euler line; the centroid is the balance point of a cardboard triangle.
Next: Exercise 5 – six proofs about centres, solved with animations ▶