It is known that
is the smallest angle in the triangle
. The points
and
divide the circumcircle of the triangle into two arcs. Let
be an interior point of the arc between
and
which does not contain
. The perpendicular bisectors of
and
meet the line
at
and
, respectively. The lines
and
meet at
.
Show that
.
Alternative formulation:
Four different points
are chosen on a circle
such that the triangle
is not right-angled. Prove that:
(a) The perpendicular bisectors of
and
meet the line
at certain points
and
respectively, and that the lines
and
meet at a certain point
(b) The length of one of the line segments
and
is the sum of the lengths of the other two.
![\angle BAC](/media/m/b/2/1/b21a9e466104c5d33646432221e142be.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![U](/media/m/d/f/a/dfa3ccb1bb2d14869d77a98d0d2baf97.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![AU](/media/m/c/b/d/cbd17cfc8e46133c11352a969e4108b4.png)
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
![W](/media/m/2/c/4/2c4aa0f61279d74f16a59bcde17578ef.png)
![BV](/media/m/5/3/6/536ec4b055e375a4f9359af5393bbaa7.png)
![CW](/media/m/b/a/d/bad87f614928c48911129949b2ac3c82.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
Show that
![AU = TB + TC](/media/m/2/5/0/250c3d7f947477827eac881781022a9f.png)
Alternative formulation:
Four different points
![A,B,C,D](/media/m/8/5/d/85d135de173dbb765c7a2f175c5b2f60.png)
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
(a) The perpendicular bisectors of
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![W](/media/m/2/c/4/2c4aa0f61279d74f16a59bcde17578ef.png)
![V,](/media/m/6/9/e/69eac8fb2d277e7a05922dd5ddadfd27.png)
![CV](/media/m/b/9/8/b98c27e266f40367d02855b467b6e882.png)
![BW](/media/m/3/9/6/39669c8ef23b163113c6a8a7c0aeafa5.png)
![T.](/media/m/f/1/b/f1b8bcb4cb384c63c9dbe37dafeb95a2.png)
(b) The length of one of the line segments
![AD, BT,](/media/m/e/0/4/e046a6275c6ca0f88a0f4218c6c3bb65.png)
![CT](/media/m/7/5/d/75d25c00070777fee6d81eb3fdcdcb2f.png)