ABCD is a convex quadrilateral with
AC and
BD
intersecting at E.
F is a point on
AD such that
EF is
parallel to CD. It is given that the circumcircle of triangle
AED is tangential to
DC. If
FD=336 and
FA=420,
what is ED?
To find the length of ED, we can use the properties of the tangent from an external point to a circle.
Let's start by analyzing the given information and the diagram provided:
- ABCD is a convex quadrilateral.
- AC and BD intersect at point E.
- F is a point on AD such that EF is parallel to CD.
- The circumcircle of triangle AED is tangential to DC.
- FD = 336 and FA = 420.
Since the circumcircle of triangle AED is tangential to DC, it means that CD is tangent to the circumcircle at point D.
Let's label the points where CD intersects the circumcircle as M and N (with M closer to D).
Now, we can make the following observations:
1. FM = DN (by the property of tangents from a common point to a circle).
2. ∠DFM = ∠DEN (opposite angles formed by the intersection of parallel lines EF and CD with transversals FD and DN).
3. ∠ADF = ∠DEN (opposite angles formed by intersecting lines AD and DN with transversal FD).
Now, using the above observations, we can set up the following proportions:
FD / FA = FM / DM (by proportionality of corresponding sides in similar triangles)
Substituting the given values:
336 / 420 = FM / DM
Simplifying:
4 / 5 = FM / DM
Since FM = DN (from observation 1), we can rewrite the equation as:
4 / 5 = DN / DM
Now, let's consider triangle DMN. We know that triangle DMN is an isosceles triangle since DM = DN (both are the lengths of tangents from D to the circumcircle of triangle AED).
In an isosceles triangle, the angle bisector is also the perpendicular bisector of the base.
Therefore, DM bisects angle MDN and is perpendicular to DN.
Now, given that ∠ADF = ∠DEN (from observation 3), we have:
∠MDN = 180° - ∠ADF
Therefore, ∠MDN = 180° - ∠ADF = 180° - ∠DEN
Since DM bisects angle MDN, we have:
∠MDN = 1/2 * (180° - ∠DEN) = 1/2 * ∠DMN
Now, using the proportion 4/5 = DN / DM, we can calculate DN:
4/5 = DN / DM
4/5 = DN / (2 * DN) (using the angle bisector property)
4/5 = 1 / 2
DN = 10
Since DN = FM, we also have FM = 10.
Now, we can find the length ED by considering triangle NED:
NE + ED = ND
NE + ED = FD + FM
NE + ED = 336 + 10
NE + ED = 346
We know that FA = 420, so AD = FA - FD = 420 - 336 = 84.
Since NE + ED = AD, we can substitute the values:
ED + NE = AD
ED + NE = 84
ED + 346 = 84
ED = 84 - 346
ED = -262
Therefore, the length of ED is -262.