Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Double perspective triangles"
(→‎Two triangles in double perspective are in triple perspective)
 
(4 intermediate revisions by the same user not shown)
Line 2: Line 2:
  
 
==Two triangles in double perspective are in triple perspective==
 
==Two triangles in double perspective are in triple perspective==
−
 
+
[[File:Exeter B.png|500px|right]]
 +
[[File:Exeter C.png|500px|right]]
 
Let <math>\triangle ABC</math> and <math>\triangle DEF</math> be in double perspective, which means that triples of lines <math>AF, BD, CE</math> and <math>AD, BE, CF</math> are concurrent. Prove that lines <math>AE, BF,</math> and <math>CD</math> are concurrent (the triangles are in triple perspective).
 
Let <math>\triangle ABC</math> and <math>\triangle DEF</math> be in double perspective, which means that triples of lines <math>AF, BD, CE</math> and <math>AD, BE, CF</math> are concurrent. Prove that lines <math>AE, BF,</math> and <math>CD</math> are concurrent (the triangles are in triple perspective).
  
Line 9: Line 10:
 
Denote <math>G = AF \cap BE.</math>
 
Denote <math>G = AF \cap BE.</math>
  
−
It is known that there is projective transformation that maps any quadrungle into square. We use this transformation for <math>BDFG</math>. We use the claim and get the result: lines <math>AE, BF,</math> and <math>CD</math> are concurrent.
+
It is known that there is projective transformation that maps any quadrungle into square.
 +
 
 +
We use this transformation for <math>BDFG</math>.
 +
We use the <i><b>Claim for square</b></i> and get the result: lines <math>AE, BF,</math> and <math>CD</math> are concurrent.
 +
 
 +
<i><b>Claim for square</b></i>
 +
 
 +
Let <math>ADBG</math> be the square, let <math>CEGF</math> be the rectangle, <math>A \in FG, G \in BE.</math>
 +
 +
Prove that lines <math>BF, CD,</math> and <math>AE</math> are concurrent.
 +
 
 +
<i><b>Proof</b></i>
 +
 
 +
Let <math>BG = a, GE = b, AF = c, A = (0,0).</math> Then
 +
<cmath>B = (-a, -a), F = (0,c), BF: y = x (1 + \frac {c}{a})+c.</cmath>
 +
<cmath>E=(b, -a),  AE: y = -\frac {a}{b}x.</cmath>
 +
<cmath>D = (-a,0), C= (b,c), CD: y = c \frac {x+a}{a+b}.</cmath>
 +
<cmath>X = CD \cap AE \cap BF = (-bk, ak),</cmath>
 +
where <math>k=  \frac {c} {a+b +{\frac {bc}{a}}}</math> as desired.
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''

Latest revision as of 13:51, 5 December 2022

Double perspective triangles

Two triangles in double perspective are in triple perspective

Exeter B.png
Exeter C.png

Let $\triangle ABC$ and $\triangle DEF$ be in double perspective, which means that triples of lines $AF, BD, CE$ and $AD, BE, CF$ are concurrent. Prove that lines $AE, BF,$ and $CD$ are concurrent (the triangles are in triple perspective).

Proof

Denote $G = AF \cap BE.$

It is known that there is projective transformation that maps any quadrungle into square.

We use this transformation for $BDFG$. We use the Claim for square and get the result: lines $AE, BF,$ and $CD$ are concurrent.

Claim for square

Let $ADBG$ be the square, let $CEGF$ be the rectangle, $A \in FG, G \in BE.$

Prove that lines $BF, CD,$ and $AE$ are concurrent.

Proof

Let $BG = a, GE = b, AF = c, A = (0,0).$ Then \[B = (-a, -a), F = (0,c), BF: y = x (1 + \frac {c}{a})+c.\] \[E=(b, -a), AE: y = -\frac {a}{b}x.\] \[D = (-a,0), C= (b,c), CD: y = c \frac {x+a}{a+b}.\] \[X = CD \cap AE \cap BF = (-bk, ak),\] where $k= \frac {c} {a+b +{\frac {bc}{a}}}$ as desired.

vladimir.shelomovskii@gmail.com, vvsss

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Double_perspective_triangles&oldid=183634"