Outside of circle Γ, point A is chosen. Tangents AB and AC to Γ are drawn, with B,C on the circumference of Γ. On the extension of AB, D is a point such that B is between A and D, and ∠ADC=25∘. The circumcircle of ADC intersects Γ again at E (different from point C ). F is the foot of the perpendicular from B to DC. What is the measure (in degrees) of ∠DEF?
To find the measure of angle DEF, we will need to use the properties of angles formed by intersecting chords and tangents of a circle.
Let's break down the problem step by step:
Step 1: Identify the relevant angles and their relationships
- ∠ADC = 25° (Given)
- ∠FBA = 90° (The foot of the perpendicular from B to DC)
- ∠BAC = ∠BCA (Tangents drawn from a point outside a circle are equal in measure)
- ∠BCD = ∠BAC (Corresponding angles)
Step 2: Find the measure of angles BAC and BCD
Since ∠BAC = ∠BCA, and ∠BCD = ∠BAC, we can assign a variable to ∠BAC for ease of calculation. Let's say ∠BAC = x.
Since the sum of angles in a triangle is 180°, we can calculate ∠BCD as follows:
∠BCD = 180° - ∠BAC - ∠BAC
∠BCD = 180° - x - x
∠BCD = 180° - 2x
Step 3: Find the measure of angles BED and BEC
Since AECD is a cyclic quadrilateral (circumcircle of ADC intersects Γ at E), the opposite angles are supplementary. Therefore,
∠BED = 180° - ∠ADC
∠BED = 180° - 25°
∠BED = 155°
Since ∠BED and ∠BEC are opposite angles, ∠BEC = ∠BED = 155°.
Step 4: Find the measure of angle DEF
∠DEF is an exterior angle of triangle BEC, so:
∠DEF = ∠BEС - ∠BDC
∠DEF = 155° - ∠BDC
To find the measure of ∠BDC, we need to use the fact that the opposite angles in a cyclic quadrilateral are supplementary to calculate ∠BDC. Since AECD is a cyclic quadrilateral, we have:
∠BDC = 180° - ∠AEC
Step 5: Find the measure of ∠AEC
∠AEC is an inscribed angle that intercepts the same arc as ∠ADC (which is 25°), so ∠AEC = ∠ADC = 25°.
Step 6: Calculate ∠BDC
∠BDC = 180° - ∠AEC
∠BDC = 180° - 25°
∠BDC = 155°
Step 7: Calculate ∠DEF
∠DEF = ∠BEС - ∠BDC
∠DEF = 155° - 155°
∠DEF = 0°
Therefore, the measure of angle DEF is 0°.