Let
be a cyclic quadrilateral with side-lengths
,
,
, and
. Let
and
be the diagonals. Then
bisects
if and only if
, and
bisects
if and only if
.
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com AB=a](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b31b2c17ef490824b6376c947bb06632_l3.png)
![Rendered by QuickLaTeX.com BC=b](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2e326df6d8734d293bac845aa63dab9a_l3.png)
![Rendered by QuickLaTeX.com CD=c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-de1a798b4cdf730b750dbe8c20227d57_l3.png)
![Rendered by QuickLaTeX.com DA=d](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-bc455b4b522444255c9876162502c41e_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com BD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-95329c4b4f11085d9454b642b291a0bb_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com BD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-95329c4b4f11085d9454b642b291a0bb_l3.png)
![Rendered by QuickLaTeX.com ab=cd](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-25b4140fc688f16870beca0259897321_l3.png)
![Rendered by QuickLaTeX.com BD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-95329c4b4f11085d9454b642b291a0bb_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com ad=bc](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-082caad8018abafbe577642fa7b7ebcd_l3.png)
Let
be a convex quadrilateral whose diagonals intersect at
. If
is cyclic, PROVE that
and
.
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com E](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ee84f23ec5accbe2212951a450e63b6d_l3.png)
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com \frac{AE}{EC}=\frac{ad}{bc}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-e9a8383d891ba305bc69d41d2372d2a4_l3.png)
![Rendered by QuickLaTeX.com \frac{BE}{ED}=\frac{ab}{cd}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-33395bddcc4afcf42d6b6edf3017fda0_l3.png)
Use similar triangles.
Let
be a convex quadrilateral whose diagonals intersect at
. If
and
, PROVE that
is cyclic.
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com E](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ee84f23ec5accbe2212951a450e63b6d_l3.png)
![Rendered by QuickLaTeX.com \frac{AE}{EC}=\frac{ad}{bc}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-e9a8383d891ba305bc69d41d2372d2a4_l3.png)
![Rendered by QuickLaTeX.com \frac{BE}{ED}=\frac{ab}{cd}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-33395bddcc4afcf42d6b6edf3017fda0_l3.png)
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
This is the converse of the previous statement in example 1. One proof of it uses similar triangles and the law of sines.
Let
be a convex cyclic quadrilateral whose diagonals intersect at
. PROVE that diagonal
bisects diagonal
if, and only if,
.
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com E](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ee84f23ec5accbe2212951a450e63b6d_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com BD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-95329c4b4f11085d9454b642b291a0bb_l3.png)
![Rendered by QuickLaTeX.com ab=cd](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-25b4140fc688f16870beca0259897321_l3.png)
By example 1 we have that . If
bisects
, then
. And so
. Similarly for the converse.
Let
be a convex cyclic quadrilateral whose diagonals intersect at
. PROVE that diagonal
bisects diagonal
if, and only if,
.
![Rendered by QuickLaTeX.com ABCD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-563addbbd3bac1a4b850c391dabc7878_l3.png)
![Rendered by QuickLaTeX.com E](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ee84f23ec5accbe2212951a450e63b6d_l3.png)
![Rendered by QuickLaTeX.com BD](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-95329c4b4f11085d9454b642b291a0bb_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com ad=bc](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-082caad8018abafbe577642fa7b7ebcd_l3.png)
By example 1 we have that . If
bisects
, then
. And so
. Similarly for the converse.
Among convex cyclic quadrilaterals, PROVE that only rectangles have the two diagonals bisecting each other.
This follows from example 3 and example 4 above, since and
yield
and
.
Brief relief
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Takeaway
Let be a convex cyclic quadrilateral. Then the following statements are equivalent:
is a rectangle
- diagonals
and
bisect each other.
Task
- (Easy verification) Let
be a convex cyclic quadrilateral with vertices at
,
,
,
. Verify that:
- diagonal
bisects diagonal
.
- diagonal