I'm currently studying to be a secondary maths teacher in Queensland, Australia, and am looking for a solid resource for filling in a gap in my mathematical knowledge base. I went through the pre-2020 senior system and, as a result, geometric proof was not covered in any form during my senior schooling years. Additionally and rather disappointingly, as best I can tell, none of the discipline-specific subjects in my teaching degree cover this area, which I'd hoped would be an opportunity to patch up this gap in my knowledge.
The current syllabus for Specialist Mathematics contains, in Unit 2 Topic 3, a section on geometric proofs, quoted below:
>Topic 3: Circle and geometric proof
Sub-topic: Circle properties and their proofs
* Prove the circle properties
- the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc
- an angle in a semicircle is a right angle
- angles at the circumference of a circle subtended by the same arc are equal
- the alternate segment theorem
- the opposite angles of a cyclic quadrilateral are supplementary and its converse
- a tangent drawn to a circle is perpendicular to the radius at the point of contact and its converse.
* Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above.
>Sub-topic: Geometric proofs using vectors
* Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus.
* Prove midpoints of the sides of a quadrilateral join to form a parallelogram.
* Prove the sum of the squares of the lengths of a parallelogram’s diagonals is equal to the sum of the squares of the lengths of the sides.
* Prove an angle in a semicircle is a right angle.
So, in the interests of being prepared to one day teach this content, I was hoping to find a good resource for teaching myself the foundations of geometric proof, particularly covering the aforementioned syllabus points. Ideally it would take me through to around a first or second year undergraduate level, since that's where the rest of my mathematical understanding sits and it positions me to both teach with an eye towards "what next" and have the capacity to answer questions that go beyond the syllabus content.
My strong preference would be a single textbook, or a pair of complementary textbooks where one starts at a high school level and the other continues into undergrad territory. I'm prepared to use audiovisual content if there are no good texts in print that do the job, though I'm assuming (and hoping) that that won't be the case.
Thanks in advance to those who offer recommendations.