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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Partly Circles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem brings together some key ideas including the value of similar triangles, employing useful lines, some algebra and that problems can be solved even when information seems limited. The problem can be used to make sense of other people's mathematical arguments. The aim of bringing the three problems together is to encourage discussion about their links and connectedness and therefore investigate relationships between problems and problem-solving experiences.

Possible approach

This printable worksheet may be useful: Partly Circles.

The first problem is designed to draw attention to similar triangles and to see these learners need to add lines.
Present the problem and give time for reflection and then discussion. If ideas are not forthcoming suggest the addition of two lines (you might add them to the diagram without discussion) and then use of angles in the same segment. If necessary spend some time adding and removing pairs of lines talking about what is known about the triangles created:
  • They are right-angled
  • They have equal angles
  • They are similar
Consider other ways of writing the result or substitute integers for three of the letters asking what the fourth must be (if a=$3$, b=$5$ and c=$6$ what must d be?)

The next two problems can be tackled in any order and the accompanying Word 2003 file has images that might stimulate additional discussion if needed.

Once solutions for all three problems are available spend time discussing connections learners can make and those that are worth drawing particular attention to.

Key questions

  • What lines can you draw that highlight additional properties
  • Can you identify similar triangles?
  • Why don't you need to worry about $ \pi$?
  • What do the first and scond problems have in common?
  • How about the second and last problems?
  • Can you think of other problems where you have used similar ideas?

Possible extension

Allow learners time to work on the problems with limited support and focus on what they see as "key moments" that helped them to solving the problems.

Possible support

Use some, or all, of the guidance sheets to help learners in structuring their own solutions using the images and statements on offer. Then focus on what is the same and what is different about each of the three problems.

Try Semi-circles

Or Bull's Eye