Random quick geo lemmas
by Inconsistent, Sep 26, 2021, 1:22 AM
Ever see the condition come up 
? Not sure what it means? Well one way to interpret it is that the center of spiral similarity from the segment 
to 
is the point 
on the circumcircle of 
such that 
. This comes from 
after 
.
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(0cm); unitsize(30);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
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draw(circle((-2.0119933619256116,2.425966215820826), 3.2785300593657425), linewidth(1));
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draw((1.068153770457503,1.302821971728902)--(3.22,-0.23), linewidth(1));
draw((2.1957050013407637,0.608888918098619)--(2.092448769116741,0.46393305363028287), linewidth(1));
draw((-6.34,-0.21)--(-1.5476747899503867,5.671450403716929), linewidth(1));
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draw((-3.989401335176457,2.815679202551603)--(-3.851431543098703,2.7032586145310584), linewidth(1));
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draw((0.9286825705007941,2.747804915090992)--(0.7902436091446904,2.6359625880608952), linewidth(1));
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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[/asy]](//latex.artofproblemsolving.com/a/8/d/a8d184be3398e0b8d3e9db876f689f27a8240b16.png)
Here's another random one: If cevians
concur, then sometimes we will wonder about properties of 
on the sides of 
respectively. We know 
swap under the nice inversion at the harmonic conjugate of 
on when the concurrence is due to Miquel (meaning 
). Here's another one: if 
, then 
is a harmonic bundle. This is by projecting the ceva-menelaus on via 
onto 
, then projecting via 
onto 
.
Another thing that isn't really a lemma, when there is an arbitrary point on that is the sole degree of freedom beyond the triangle, it might be a good idea, in general, to consider its projection via onto when experimenting for similar triangles and cyclic quadrilaterals.
? Not sure what it means? Well one way to interpret it is that the center of spiral similarity from the segment 
to 
is the point 
on the circumcircle of 
such that 
. This comes from 
after 
.![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(0cm); unitsize(30);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -10.457861904816939, xmax = 4.966382807294912, ymin = -4.336543019812374, ymax = 6.20830120164101; /* image dimensions */
pen qqwuqq = rgb(0,0.39215686274509803,0);
draw(arc((-6.34,-0.21),0.4449301359263034,50.82615299044904,66.42867268465027)--(-6.34,-0.21)--cycle, linewidth(1) + qqwuqq);
draw(arc((3.22,-0.23),0.4449301359263034,128.93411606425818,144.53663575845943)--(3.22,-0.23)--cycle, linewidth(1) + qqwuqq);
/* draw figures */
draw(circle((-1.5578941209700194,0.7866101763305489), 4.884850917051483), linewidth(1));
draw((-6.34,-0.21)--(3.22,-0.23), linewidth(1));
draw(circle((-2.0119933619256116,2.425966215820826), 3.2785300593657425), linewidth(1));
draw((-5.283503068252974,2.2115283656856604)--(-6.34,-0.21), linewidth(1));
draw((-5.730190237421473,0.965179524434851)--(-5.8933128308315,1.0363488412508102), linewidth(1));
draw((-5.283503068252974,2.2115283656856604)--(-4.08,4.97), linewidth(1));
draw((-4.08,4.97)--(1.068153770457503,1.302821971728902), linewidth(1));
draw((1.068153770457503,1.302821971728902)--(3.22,-0.23), linewidth(1));
draw((2.1957050013407637,0.608888918098619)--(2.092448769116741,0.46393305363028287), linewidth(1));
draw((-6.34,-0.21)--(-1.5476747899503867,5.671450403716929), linewidth(1));
draw((-4.036243246851684,2.7581917891858714)--(-3.8982734547739297,2.645771201165327), linewidth(1));
draw((-3.989401335176457,2.815679202551603)--(-3.851431543098703,2.7032586145310584), linewidth(1));
draw((-1.5476747899503867,5.671450403716929)--(3.22,-0.23), linewidth(1));
draw((0.8820816009049207,2.8054878156560346)--(0.7436426395488169,2.6936454886259376), linewidth(1));
draw((0.9286825705007941,2.747804915090992)--(0.7902436091446904,2.6359625880608952), linewidth(1));
draw((-5.283503068252974,2.2115283656856604)--(-1.5476747899503867,5.671450403716929), linewidth(1));
draw((1.068153770457503,1.302821971728902)--(-1.5476747899503867,5.671450403716929), linewidth(1));
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dot((-4.08,4.97),dotstyle);
label("$A$", (-4.021205938416417,5.125637870887006), NW * labelscalefactor);
dot((-6.34,-0.21),dotstyle);
label("$B$", (-6.275518627109687,-0.06521371491986393), SW * 2);
dot((3.22,-0.23),dotstyle);
label("$C$", (3.275648290774959,-0.0800447194507407), SE * labelscalefactor);
dot((-5.283503068252974,2.2115283656856604),dotstyle);
label("$D$", (-5.222517305417436,2.3670710281439264), SW *2);
dot((1.068153770457503,1.302821971728902),linewidth(4pt) + dotstyle);
label("$E$", (1.1251526337978262,1.4178867381678133), E * labelscalefactor);
dot((-1.5476747899503867,5.671450403716929),linewidth(4pt) + dotstyle);
label("$H$", (-1.4851041636364872,5.793033074776461), N * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/a/8/d/a8d184be3398e0b8d3e9db876f689f27a8240b16.png)
Here's another random one: If cevians
concur, then sometimes we will wonder about properties of 
on the sides of 
respectively. We know 
swap under the nice inversion at the harmonic conjugate of 
on when the concurrence is due to Miquel (meaning 
). Here's another one: if 
, then 
is a harmonic bundle. This is by projecting the ceva-menelaus on via 
onto 
, then projecting via 
onto 
.Another thing that isn't really a lemma, when there is an arbitrary point on
that is the sole degree of freedom beyond the triangle, it might be a good idea, in general, to consider its projection via onto when experimenting for similar triangles and cyclic quadrilaterals.
October 2023
July 2023
Reason: edit