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| Where is the centre of a triangle? | |
| There are many candidates for the centre of a triangle, such as circumcentre, orthocentre, incentre and centre of gravity. Construct each of these. Which one seems to be the best candidate? | the circumcentre the orthocentre the incentre the centre of gravity |
| When you have constructed them all, have a look at Euler's line, which compares all four points. | Euler's line |
| Elegant triangles from triangles. | triangles from triangles |
| Investigate Ceva's theorem | Ceva's theorem more Ceva's theorem |
| The nine-point circle | |
| The altitudes of a triangle meet at H. Construct the midpoints, A', B', C', of each side of the triangle; the midpoints, A'', B'' and C'', of the segments AH, BH and CH; and the feet, D, E and F, of the altitudes. These nine points form a circle. | nine-point circle |
| Touching circles | |
| Construct the circle that touches the two given circles. | touching circles 1 touching circles 2 |
| Circles and tangents | |
| Two circles will have two tangents in common. Can you construct them? And what happens with three circles? | circles and tangents |
| Cyclic and tangential polygons | quadrilaterals hexagons |
| Circles and triangles | |
| Construct four straight lines. These will form four triangles. Construct the circumcircle and circumcentre of each triangle. What property do these four centres have? | four straight lines |
| Four points on a circle create four overlapping triangles. Construct the incentres of these four triangles and discover a property that these points have. | four incentres |
| Construct the lines that pass through the intersections of three circles - taken in pairs. | three circles |
| A point moves round a circle. | a constant |
| A triangle and three circles - the pivot theorem | the pivot theorem |
| Construct three equal line segments from one angle. | Polya's problem |
