Geo with angle bisector

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Source: DGO 2021, Individual stage, Day2 P1

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#1 h Dec 26, 2021, 7:12 PM • 2 Y

Y by Noob_at_math_69_level, ImSh95

Let $M$ be an interior point inside a triangle $ABC$ such that $\angle MAB < \angle MAC$ . Is it always true that
$\frac{MB}{MC}<\frac{AB}{AC}$ ?

Proporsed by wassupevery1

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#2 h Dec 26, 2021, 7:13 PM • 1 Y

Y by ImSh95

Sorry for the unrelated question, but what's DGO? Otherwise, for the question, probably law of sine helps.

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#3 h Dec 26, 2021, 7:25 PM • 2 Y

Y by Noob_at_math_69_level, ImSh95

a_507_bc wrote:

Sorry for the unrelated question, but what's DGO? Otherwise, for the question, probably law of sine helps.

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#4 h Dec 26, 2021, 7:55 PM • 1 Y

Y by ImSh95

$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); 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 = -11.92, xmax = 15.4, ymin = -7.32, ymax = 6.98; /* image dimensions */ pen qqwuqq = rgb(0,0.39215686274509803,0); draw(arc((0.16,1.14),0.6,-160.76606420920794,-120.97284232859343)--(0.16,1.14)--cycle, linewidth(0.6) + qqwuqq); draw(arc((0.16,1.14),0.6,-120.97284232859343,-81.17962044797893)--(0.16,1.14)--cycle, linewidth(0.6) + qqwuqq); draw(arc((4.6,3.9),0.6,217.78184207622843,231.23032461400564)--(4.6,3.9)--cycle, linewidth(0.6) + qqwuqq); draw(arc((4.6,3.9),0.6,204.33335953845113,217.7818420762284)--(4.6,3.9)--cycle, linewidth(0.6) + qqwuqq); /* draw figures */ draw((4.6,3.9)--(-7.12,-1.4), linewidth(0.6)); draw((-7.12,-1.4)--(0.52,-1.18), linewidth(0.6)); draw((0.52,-1.18)--(4.6,3.9), linewidth(0.6)); draw((0.52,-1.18)--(0.16,1.14), linewidth(0.6)); draw((0.16,1.14)--(-7.12,-1.4), linewidth(0.6)); draw((-2.048809148845591,-1.2539709440767053)--(4.6,3.9), linewidth(0.6)); draw((-1.2633228022644833,-1.2313522272903383)--(0.16,1.14), linewidth(0.6)); draw(arc((0.16,1.14),0.6,-160.76606420920794,-120.97284232859343), linewidth(0.6) + qqwuqq); draw(arc((0.16,1.14),0.47,-160.76606420920794,-120.97284232859343), linewidth(0.6) + qqwuqq); draw(arc((0.16,1.14),0.6,-120.97284232859343,-81.17962044797893), linewidth(0.6) + qqwuqq); draw(arc((0.16,1.14),0.47,-120.97284232859343,-81.17962044797893), linewidth(0.6) + qqwuqq); draw(arc((4.6,3.9),0.6,217.78184207622843,231.23032461400564), linewidth(0.6) + qqwuqq); draw((4.225582586081088,3.5319828806233673)--(4.118606182104256,3.4268351322300434), linewidth(0.6) + qqwuqq); draw(arc((4.6,3.9),0.6,204.33335953845113,217.7818420762284), linewidth(0.6) + qqwuqq); draw((4.150259228996044,3.6291527388051197)--(4.0217618658520555,3.5517678070351537), linewidth(0.6) + qqwuqq); /* dots and labels */ dot((4.6,3.9),dotstyle); label("$A$", (4.68,4.1), NE * labelscalefactor); dot((-7.12,-1.4),dotstyle); label("$B$", (-7.42,-1.78), NE * labelscalefactor); dot((0.52,-1.18),dotstyle); label("$C$", (0.5,-1.58), NE * labelscalefactor); dot((0.16,1.14),dotstyle); label("$M$", (0.24,1.34), NE * labelscalefactor); dot((-2.048809148845591,-1.2539709440767053),linewidth(4pt) + dotstyle); label("$D$", (-2.28,-1.66), NE * labelscalefactor); dot((-1.2633228022644833,-1.2313522272903383),linewidth(4pt) + dotstyle); label("$E$", (-1.36,-1.66), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
The anser is NO.
From example that I noted @above we get $\frac{MB}{MC}=\frac{BE}{EC}>\frac{BD}{DC}=\frac{AB}{AC}$ so we are done.

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#5 h Dec 28, 2021, 11:35 AM • 1 Y

Y by ImSh95

Sketch:- Counterexample:- Let $AB$ > $AC$ & $D$ be the intersection of $BC$ & $A$ angle bisector,any point $M$ within ∆ $ABD$ and $A$ -apollonius circle works.Let $E$ be intersection of $M$ angle bisector with $BC$ .Since $M$ is within $A$ apollonius circle and $M$ apollonius circle & $A$ apollonius circle don't intersect (as their radax is perp. bisector of $BC$ ), $E$ lies within $D$ & $C$ .So, done.

This post has been edited 7 times. Last edited by GeoKing, Dec 28, 2021, 12:05 PM
Reason: :(((((((((((((((((((((((((((((((((

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