Where is the centre of a triangle?
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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?
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 The
circumcentre
The orthocentre
The
incentre
The
centre of gravity
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When you have constructed them all, have a look at Euler's
line, which compares all four points.
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Euler's
line
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Elegant triangles from triangles.
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Triangles
from triangles
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Investigate Ceva's theorem.
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Ceva's
theorem
More Ceva's theorem
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| The nine-point circle |
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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.
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Nine-point
circle
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| Circles and triangles |
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Construct four straight lines. These will form four triangles.
Construct the circumcircle and circumcentre of each triangle.
What property do these four centres have?
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Four
straight lines
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Four points on a circle create four overlapping triangles.
Construct the incentres of these four triangles and discover
a property that these points have.
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Four
incentres
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Construct the lines that pass through the intersections of
three circles - taken in pairs.
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Three
circles
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A point moves round a circle.
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A
constant
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A triangle and three circles - the pivot theorem.
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The
pivot theorem
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