324. The Four Triangle Hearts · Where medians, bisectors, and altitudes meet

Try to break it

Drag A, B, or C. The three medians always meet at G (centroid); the three angle bisectors at I (incentre); the three perpendicular bisectors at O (circumcentre); the three altitudes at H (orthocentre). Try to drag a vertex so any of those line-triples fail to concur — impossible. Each set is always concurrent.

How you build it

Construct the four triangle centres.

• Place point A as the first vertex of the triangle.
• Place point B as the second vertex of the triangle.
• Place point C as the third vertex of the triangle.
• Draw segment AB to form one side of the triangle.
• Draw segment BC to form one side of the triangle.
• Draw segment CA to complete triangle ABC.
• Mark the midpoint D of side BC.
• Mark the midpoint E of side CA.
• Mark the midpoint F of side AB.
• Draw the median AD from vertex A to midpoint D.
• Draw the median BE from vertex B to midpoint E.
• Draw the median CF from vertex C to midpoint F.
• Mark point G where the three medians intersect. This is the centroid.
• Put the compass point on A and draw an arc that crosses AB and AC. (This sets the radius the next arcs reuse.)
• Place the compass on the point where the arc crossed AB and draw an arc inside the angle (same radius, one click).
• Place the compass on the point where the arc crossed AC and draw an arc. It meets the previous arc at K.
• Draw the line through A and K — the bisector of angle A.
• Put the compass on B and draw an arc that crosses BA and BC.
• Place the compass on the BA cut point and draw an arc inside angle B.
• Place the compass on the BC cut point and draw an arc. It meets the previous arc at L.
• Draw the line through B and L — the bisector of angle B.
• Mark point I where the two angle bisectors intersect. This is the incentre.
• At midpoint D, erect the perpendicular to BC — the perpendicular bisector of BC.
• At midpoint E, erect the perpendicular to CA — the perpendicular bisector of CA.
• Mark point O where the two perpendicular bisectors intersect. This is the circumcentre.
• From vertex A, drop the perpendicular to side BC — the altitude from A.
• From vertex B, drop the perpendicular to side CA — the altitude from B.
• Mark point H where the two altitudes intersect. This is the orthocentre.

The proof, step by step

Prove that each of the four triangle centres is a point of concurrency.

• The three medians of a triangle are concurrent. Their intersection is the centroid (G).
• The centroid divides each median in the ratio 2:1.
• The three angle bisectors are concurrent. Their intersection is the incentre (I).
• The incentre is equidistant from all three sides.
• The three perpendicular bisectors of the sides are concurrent. Their intersection is the circumcentre (O).
• The circumcentre is equidistant from all three vertices.
• The three altitudes are concurrent. Their intersection is the orthocentre (H).

Worked example

In a triangle, the point of concurrency of the perpendicular bisectors of the sides is called the:

The circumcentre is the point where the perpendicular bisectors of the sides meet. It is equidistant from all three vertices.

• Centroid
• Incentre
• Circumcentre — correct
• Orthocentre