Ratio Chasing is the habit of propagating side, area, and segment ratios through a diagram until the target ratio appears. In contest math, that habit turns a crowded setup into a relation the student can test, bound, count, or compute. MathGrit teaches it as a recognizable signal, a deliberate move, and a final translation back to the original question.
$$\text{Angle bisector: } \frac{BD}{DC} = \frac{AB}{AC}; \qquad DE \parallel BC \Rightarrow \frac{AD}{DB} = \frac{AE}{EC}$$
You propagate length and segment ratios through a figure — via parallels, bisectors, and similar triangles — until the ratio you need appears.
Why It Works
1
Parallel lines cut transversals proportionally (Thales): if $DE \parallel BC$ in $\triangle ABC$ with $D$ on $AB$, $E$ on $AC$, then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$.
2
The internal angle bisector from $A$ meets $BC$ at $D$ with $\dfrac{BD}{DC} = \dfrac{AB}{AC}$ (proof: drop a parallel and use similar triangles, or compare areas with equal heights and equal bisected angles).
3
Similar triangles supply equal ratios of all corresponding sides, letting you carry a ratio from one part of the figure to another.
4
Multiply ratios along a chain ($\tfrac{x}{y}\cdot\tfrac{y}{z} = \tfrac{x}{z}$) to relate the first and last segments directly.
Worked Examples
Example 1
In $\triangle ABC$, $AB = 8$ and $AC = 12$. The internal bisector of $\angle A$ meets $BC$ at $D$. If $BC = 15$, find $BD$.
In $\triangle ABC$, $D$ is on $BC$ with $BD:DC = 3:2$. A line through $D$ parallel to $AB$ meets $AC$ at $E$. Find $\dfrac{AE}{EC}$.
1
Line $DE \parallel AB$ inside $\triangle ABC$ (with $D$ on $BC$, $E$ on $AC$) cuts the two sides $CB$ and $CA$ proportionally from vertex $C$.
2
So $\dfrac{CE}{EA} = \dfrac{CD}{DB} = \dfrac{2}{3}$.
3
Therefore $\dfrac{AE}{EC} = \dfrac{3}{2}$.
Answer:
$\dfrac{AE}{EC} = \dfrac{3}{2}$
Olympiad / Challenge
In $\triangle ABC$, point $D$ on $BC$ has $BD:DC = 1:3$, and $E$ on $AD$ has $AE:ED = 2:1$. Line $BE$ extended meets $AC$ at $F$. Find $\dfrac{AF}{FC}$.
1
Use Menelaus on $\triangle ADC$ with the transversal line $B\text{-}E\text{-}F$ (which hits side $DC$ extended at $B$, side $AD$ at $E$, side $CA$ at $F$).
2
Menelaus (unsigned magnitude $= 1$): $\dfrac{DB}{BC}\cdot\dfrac{CF}{FA}\cdot\dfrac{AE}{ED} = 1$, taking the points $B$ on line $DC$, $F$ on line $CA$, $E$ on line $AD$.
3
Compute the factors: from $BD:DC = 1:3$, side $BC = BD + DC$ splits as $4$ parts, so $\dfrac{DB}{BC} = \dfrac{1}{4}$; and $\dfrac{AE}{ED} = 2$.
Generalization: any ratio chase multiplies a chain of local ratios into a global one. Thales (parallel), the angle-bisector theorem, similar-triangle ratios, Menelaus, and mass points are all interchangeable engines for the same job — pick the one whose hypothesis (a parallel? a bisector? a transversal?) is already present in the figure.
Where it appears: ratio chasing is the bread-and-butter of cevian and transversal problems on AMC/AIME ('a line through ... meets ... at ...; find the ratio'). It also feeds coordinates and vectors, where a ratio becomes a weighted average of position vectors.
Pitfall: orientation and which segment is the 'whole'. With a transversal hitting a side EXTENDED (Menelaus), the foot can lie outside the segment, so $\dfrac{DB}{BC}$ is a part-to-whole that needs the configuration checked. Confusing $\dfrac{DB}{DC}$ with $\dfrac{DB}{BC}$ is the classic ratio-chase error.
Spot the Signal
Look for problems where the key step is propagating side, area, and segment ratios through a diagram until the target ratio appears.
You can describe the hard part as propagating side, area, and segment ratios through a diagram until the target ratio appears, but a direct attack starts producing clutter.
The problem rewards preserving structure instead of expanding, listing, or guessing too early.
Learn the Move
Start by identify the diagram relation that calls for ratio chasing, then rewrite the givens around it.
Name the relation that makes Ratio Chasing legal before doing computation.
Use the new relation to replace the messiest part of the problem with a cleaner one.
Translate the result back to the quantity the problem actually asks for.
Avoid These Traps
Do not use Ratio Chasing just because the surface looks familiar; verify the required condition first.
Applying Ratio Chasing because it sounds relevant, without checking the trigger first.
Stopping after spotting the technique instead of finishing the calculation or proof.
MathGrit Coach Note
Let ratio chasing reveal the geometric configuration; then compute only what remains.
Try it on:
Practice a contest problem where the key step is propagating side, area, and segment ratios through a diagram until the target ratio appears.