Does the angle between Venus and the Sun max out when the former is half lit?

Question

Christiaan Huygens accurately calculated the distance from Earth to the Sun in 1659 to be 1.023 times our modern figure of 1AU=1.495978707e11 metres. They say he noticed, like Galileo before him, that Venus underwent phases just like the moon does. So, he reasoned, when Venus is half-lit, like a quarter moon, then the Sun, Venus and the Earth form a right triangle. Thus, if we knew the distance from the Earth to Venus at this moment of configuration, then we can calculate the distance from the Sun to the Earth using trigonometry. But, I need to prove that the maximum deflection of the angle between Venus and the Sun indicates Venus is half-lit, because I have to use a telescope on Earth to measure the angle and it's difficult to see enough detail. Please see the attached diagram and tell me I'm on the right track. Although the diagram is not exactly to scale, both Venus and the Earth are moving in a counter clockwise direction and Venus has a higher angular velocity than the Earth.

Answer 1

Draw a small circle at the dot labelled $V$. The line from the earth to that circle bisects it, so Venus is half-lit.

Post-comment addition The question said "I need to prove that [when venus is at maximum deflection from the sun], Venus is half-lit." But apparently you needed more than this -- you need to "prove that the angle between the sun and venus is at a maximum magnitude."

I don't think you need to prove this.

I showed that if the angle is at a maximum, then venus is half-lit. That's what you really need.

I suppose you might also need this: if the angle is at a maximum, then the triangle from earth to venus to sun has a right angle at $V$. So let's prove that (assuming the orbit of venus and the earth are both circular).

Here's a proof: Look at your picture. The line from $E$ through $V$ has the property that exactly one point of venus's orbit (labelled "V") is on that line, and all other orbit points are to the right (sun side) of it (as seen from the earth), so the sun-earth-venus angle is smaller for all those points. Hence this particular line is the one where the sun-earth-venus angle is greatest. Furthermore, it's a line containing exactly one point of a circle (venus's orbit), hence is a tangent line to that circle. And any tangent line to a circle is perpendicular to the circle-radius (the line SV in your diagram) at the point of tangency (V in your diagram). So we're done.